philatelic latin squareshome.cc.umanitoba.ca/~loly/lolystyan3-cshpm.pdf · 1947) on 9 april 2005....

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Proceedings of theCanadian Sociefy for History and Philosophy of MathematicsSoci6t6 Canadienne d'Ilistoire et de Philosophie des Math6matiquesVolume 23,2010, Pages 273-297rssN a825-5924

PHILATELIC LATIN SQUARES

PETER D. LOLY AI{D GBORGE P. H. STYAN

AesrnACT. Postage starnps are occasionally issued in sheetlets (smallsheets, miniature sheets, souvenir sheets) of n different stamps printedin an n x n affay with n copies of each of the n stamps. We focus on

sheetlets that form a phi,lateli,c Lati,n square (PLS): each of the n stampsappears exactly once in each row and exactly once in each column. We

give examples for rL - 2,3,4,5 building on results in [5, 16, L7,18, 30];

see also 129, 321.

Rnsurraf. Des timbres sont parfois 6mis en planches (petites, miniatures,souvenirs) de n diff6rents timbres imprim6s dans un tableau n x n avec

?? exemplaires de chacun des n timbres. Nous nous concentrons sur les

planches qui forment un carrti latin philat6lique (PLS en anglais) : cha-

cun des n timbres apparat exactement une fois dans chaque ligne etexactement une fois dans chaque colonne. Nous donnons des exemplespour rL - 2;3;4; 5 en nous basant sur les r6sultats de [5, L6, L7,18, 30];

voir aussi {29, 321.

1. INrRooucrIoN AND MISE-EN-ScENE

Postage stamps are occasionally issued in sheetlets (small sheets, miniaturesheets, souvenir sheets) of n different stamps printed in an n x n array withn copies of each of the n stamps. Here we focus on sheetlets where each ofthe n stamps appears exactly once in each row and exactly once in each

column. We caII such a sheetlet a phi,latelic Latin square (PLS) and have

found PLS for rL:2,3,4,5; we wouid be pleased to learn of any PLS withn ) 6. For more about Latin squares see, e.g., [6,24J.

The mathematics educator William Leonard Schaaf (1898-1992) in his

Mathemati,cs and Sc'ience: An Aduenture i,n Postage Stamps [25, p. xiii] ob-

served that "The postage stamps of the world are, in effect, a mirror of civi-lization", while Skaggs [27] refers to postage stamps as "Icons of communica-tion". Morever, "Philately is the study of stamps and is distinct from stamp

collecting-it is possible to be a philatelist without owning any stamps!" [37]and at this 2010 Annual Meeting of the CSHPM: "Mathematical philatelyis a liberal artl" r

2000 Mathernatics Subject Classi.ficati,on. 01445, 01A50,01A55, 01460, 05-03, 05499,05B15.

Key words ond, ph,rases. almost-PlS, balanced cornersr block-Latin square, block-PlS,embedded-PlS, Europa CEPT stamps series; European Conference of Postal and Telecom-

munications Administrations (CEPT), experimental design, knight's move design' KnutVik design, magic-Latin, miniature sheet, philatelic Latin square (PLS), postage stamps,rainbow-PlS, souvenir sheet, sheetlet, Sudoku form, topical philately, World Wide Fundfor Nature (WWF), World Wildlife F\rnd.

lThir pup". is based on the talk presented (by George P. H. Styan) in the SpecialSession on Mathematics and the Liberal Arts at the Annual Meeting of the CanadianSociety for the History and Philosophy of Mathematics, Concordia University, Montr6al(Qu6bec), Canada, May 31,2010, and, in part, on [5, 16, 17, 18,30].

273 @ Peter D. Loly and George P.H. Styan

;;,*-g:r+-'*-*' l:,-3rf-

We were surprised to find that Latin square designs have been used inprinting postage stamps, with the first such use apparently being in

o 1949 USSR 2 x 2: Scntt 1359-1360 (Figure 2.1,teft, panel),r 1960 T\rkey 5 x 5: Scott 1488-1492 (Figure 5.2),o lg72 Canada 4 x 4: Scott 582-585 (Figure 4.2,left panel),o 1976Isle of Man 3 x 3 : Scott 86-88 (Figure 3.2, left panel).

Throughout this paper Scottcatalog numbers axe as found in the Scott Stan-

dard Postage Stamp Catalogue [261.Latin square designs of size 4 x 4 have been used in Europe at least since

the 13th century, when the Majorcan writer and philosopher Ramon Llull(1232-1316) published the "First elemental figure" in his ,4rs Demonstratiaa

[15] in 1283. As described by Preece [23], probably the earliest use of a Latinsquare in an experimental design was by the agronomist Flangois Crett6 de

Palluel (L74L-1795), who in 1788 published a study [fl of an experimentinvolving the fattening of 16 sheep in Flance, four of each of four differentbreeds. The advantage of using a Latin-square design in this experimentwas that only 16 sheep were needed rather than 64 for a completely cross-

classified design. The purpose here was to show that one might just as wellfeed sheep on root vegetables during the winter; this was much cheaper, and

easier, than the normal diet of corn and hay.

2. PLS lurnrH n:2There are many 2 x 2 PLS but the oldest we have identified is from theUSSR (Figure 2.1, left panel) issued in 1949 in celebration of the 150thbirth anniversary of Alexander Sergeyevich Pushkin (1799-1837), who isconsidered to be the greatest Russian poet [37]. The stamp in the top-leftcorner shows Pushkin in 1822 and the stamp in the top-right corner is based

on a portrait by Orest Adamovich Kiprensky (1782-1836) painted in 1827.

This is the oldest PLS (of any size) that we have identifled to date.The 2 x 2 PLS (Figure 2.1, right panel) was issued by Romania in celebra-

tion of the centenary of the Scouting Movement. The stamp on the top-ieftof the PLS shows its founder, Robert Stephenson Smyth Baden-Powell, lstBaron Baden-Powell (1857-1941), whose father, Reverend Professor Baden

Powell (1796-1860), was Savilian Professor of Geometry at Oxford [22]. ThisPLS was issued in the "Europa" series. Europa stamps have been issued an-nually since 1956; from 1959-1992 the Europa series stamps represented theEuropean Conference of Postal and Telecommunications Administrations(CEPT2). From 1993 to date the stamps represented PostEurop {szcl, the

2Th" u".ory* CEPT comes from the Ftench: "Conf6rence europ6enne des administra-tions des postes et t6l6communications".

274

Association of European Public Postai Operators, one of the constituent

unions of the Universal Postal Union. There are many 2 x 2 PLS issued in

the Europa series and we recommend the Mclean website [20]-

(right) Romania 2A07, Scott 4941-4942-

Frcuae 2.2: Isle of Man 1981 , Scott 198-199; (right) Great Britain 2005, Scott 2279ab-

Frcuns 2.1: USSR 1949, Scott 1359-1360;

275

A 2 x 2 PLS (Figure 2.2,Left panel) was issued by the Isle of Man in cele-

bration of the marriage of The Prince Charles, Prince of Wales (b. 19a8) toLady Diana Spencer (1961-1997) on 29 July 1981 and a 2 x 2 PLS (Figure2.2, right panel) was also issued by Great Britain in celebration of his ma^r-

riage to Camilla Parker Bowles (b. 1947) on 9 April 2005. Camilla, Duchess

of Cornwall is the current holder of the titles Princess of Wales and Duchess

of Cornwall and of Rothesay [37]. The marginal inscription (in English andWelsh) states that the wedding took place on 8 April 2005, but it was de-

layed till the following day 9 April due to the Prince's attendance at theApril 8 funeral of Pope John Paul II (1920-2005).

The Isle of Man is a self-governing British Crown Dependency, located inthe Irish Sea between the islands of Great Britain and Ireland, within theBritish Isles. The head of state is Queen Elizabeth II, who holds the title ofLord of Mann (sic). The island is not part of the United Kingdom, but itsforeign relations and defence are the responsibility of the UK Government.IsIe of Man stamps are indexedby Scott [26] under Great Britain.

2.1. 2x2 embedded-PlS.

FrcuRe 2.3: Canada 2009, Scott 2326-2327.

276

There are many 2 x 2 PLS in sheetlets of 4 stamps but it seems there are even

more 2 x 2 PLS embedded in sheetlets of 4k stamps, arranged "checkerboard-style", with k ) 2. We will refer to such sheetlets as embedded-PlS. Forexample, in the PLS issued by Canada in a series to "Preserve the PolarRegions and Glaciers" (Figure 2.3), k:4 and embedded is a2 x 2 PLS

featuring a stamp with a polar bear (Ursus maritimus) on the main diagonaland a 2 x 2 PLS featuring a stamp with an Arctic Tern {Sterna paradisaea)

on the main diagonal. There are in all in this sheetlet 5 copies of the PLSfeaturing the polar bear and 4 copies of the PLS featuring the arctic tern.

In Figure 2.4 we display two sheetlets issued for the World Wide Fund forNature (WWF), an international non-governmental organization workingon questions regarding the conservation, research and restoration of theenvironment, formerly called the World Wildlife F\rnd, which remains itsofficial name in the United States and Canada. There are many PLS issued

for the WWF and we recommend the websites of Groth AG [13] and Steurs

[28].

FIGURE 2.4:Yiet Nam 2000, Scott2966-2969; (righi) Turkmenistan (Mary Velayat) 1998.

The Viet Nam sheetlet (Figure Z.4,left panel) features the Saola or Vu

Quang ox (Pseudoryr ngheti,nhensis), one of the world's rarest mammals. Itis a forest-dwelling bovine found only in Viet Nam and in La,os, near theViet Nam-Laotian border. Rows 2 and 3 form a2 x2PLS.

The other sheetlet (Figure 2.4, right panel) was issued by T[rrkmenistan(Mary Velayat) 1998 and features "wild animals and birds". Ma.y Velayat is

the southeastern region of T\rrkmenistan on the T\-uan Lowland, within the

277

+ : *;_" r,:qi.I '";- 1 .-+:u'FT

,t:

:

limits of the Karakum desert. The top two lows, and the bottom two rows,each form two 2 x 2 PLS which are "block-reflections" of each other, andso embedded in this sheetlet are four 2 x 2 PLS. For stamps from countriesand certain regions that were part of the ussR we recommend Bistamp [2].

2.2. 2x2 block-PlS.

FrcuRs 2.5: Macau 2008 , Scott 1248ah. (Joint issue with Singapore 2008.)

Frcune 2.6: Singapore 2008, Scott 1318ad-1319ad. (Joint issue with Macau 2008.)

{

278

There is a2x2 block-PlS entitled "Local gastronomy" from Macau (Figure2.5), issued jointly with Singapore (entitled "Local delights"), where each

block comprises 4 stamps, and so, 8 distinct stamps in all in a sheetlet of16 stamps. The Latin squaxe underlying a 2x2 block-PlS is a "block-Latinsquare". We do not know if the Singapore stamps (Figure 2.6) were also

issued as a 2x2 block-PlS and we welcome information about this.Featured on the stamps are 4 Macau dishes: Carne de porco d Alentejana

(Pork and clams), Lombo de bacalhau braseado em lascas (Grilled salt cod

slices), Yangzhou fried rice, Crispy fried chicken and 4 Singapore dishes:

Roti prata (fried flour-based pancake), Hainanese chicken rice, Laksa (Spicynoodle soup), Satay (Meat marinated in a spicy peanut sauce on a skewer).

FtcuRe 2.7: Hong Kong 2003, Scott 1063-1068.

279

The only other 2 x 2 block-PlS we have found is from Hong Kong (Fig-we 2.7) issued for "World heritage in China". Here each block comprises 3

stamps with the two blocks in the top "row-block" (of 6 stamps) repeatedin the bottom row-block; the sheetlet, therefore, includes 2 distinct 2 x 2

block-PlS (block-reflections of each other) one being the top 2 rows andthe other being the bottom 2 rows. The stamps feature (HK$1.40) PotalaPalace, ($1.80) Imperial Palace of the Hing and Qing Dynasties, ($2.40)Mausoleum of the First Qin Emperor, ($2.50) Mount Huangshan, ($3.00)

Old Town of Lijiang, ($5.00) Jiuzhaigou Valley.

3. PLS wITH n:3There,are just two types of 3 x3 PLS: one-step backwards circulant (C1B) asillustrated by the sheetlet depicting trees of the Holy Land issued by Israel(Figure 3.1, left panel); depicted are the Greek strawberry tree (Arbutusandrachne), Judas tree {Cerci,s si.liquastrame), and Vallonea oak (Quercusi,thaburensis). This PLS (Figure 3.1, right panel) is tirg one-step forwardscirculant (ClF) issued by the "Grenadines of St. Vincent" in 1993 for ElvisPresley (1935-1977); a PLS with the same design was also issued in 1gg3by Dominica, Gambia., Maldives, Nevis, and St. Vincent {Scott 1767). As ofOctober 31, 2010, we have identified 45 (including 18 from Abkhazia) PLSwith n : 3: only 8 of type C1B and the other 37 of type C1F, including 6 forElvis Presley a^nd 9 embedded (6 from Costa Rica and 3 from St. Vincent).

FlcuRs 3.1: Israel(right) "Grenadines of St. Vincent" 1993, Scott

1981 ,, Scatt 798-800;1347ac (listed under "St. Vincent Grenadines").

28A

Saint Vincent and the Grenadines is a nation which lies at the southernend of the eastern border of the Caribbean Sea where it meets the AtlanticOcean. Its 389 km2 (tSO mi2) territory consists of the main island of SaintVincent and the northern two-thirds of the Grenadines, which are a chainof smaller islands stretching south from Saint Vincent Island to Grenada.

The oldest 3 x 3 PLS that we have identified is from the Isle of Man issued

in 1976. Issued for the Europa CEPT prograrnme, this 3 x 3 PLS (Figure3.2, left panel) is one of a pair featuring Manx ceramic art: (top row, left toright) Barrosse beaker (Bronze Age), souvenir teapot (3-legged man, 19th

century), Laxey jug (1854). Another pair of 3 x 3 PLS were issued in 1978

featuring carved gravestones: (Figure 3.2, right panel, top row, left to right)wheel-headed cross-slab, Celtic wheel-cross, Keeil Chiggyrt stone. All four3 x 3 PLS issued by the Isle of Man h 1976/1978 are of the one-step forward.s

circulant (ClF) type.

FIcURE 3.2: Isle of Man 1976, ,Scotr 86-88 & (right) 1978, Scott 131-133.

3.1. 3x3 embedded-Pls. Three distinct 3 x 3 PLS are embedded in the4 x 4 sheetlet (Figure 3.3) issued by "St. Vincent & The Grenadines" forSandy Koufax (b. Sanford Braun, 1935), the American left-handed formerbaseball pitcher in US Major League Baseball who played his entire careerfor the Brooklyn/Los Angeles Dodgers, from 1955-1966. All four 3 x 3 PLSare of the one-step forwards circulant (C1F) type with the PLS formed fromthe first 3 rows and columns repeated in the PLS formed from the last 3 rowsand columns. We have also found six 3 x 3 PLS embedded in sheetlets fromCosta Rica issued in 2005 {Scott 586ac, 596ac) depicting flora and fauna innational parks (3) and indigenous fruits (3).

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Mt Ii{,ffii.i =l l*i-ffi$ffii iiffi.ilffi}:,itiitit7*+ii,,',.' il:io,: I I ir1J.,.... -. .. -.-;. ... -

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281

--rc-

FIGURE 3.3: "St, Vincent & The Grenadines" 1996, Scott2356ac (lGted under "St. Vincent").

4. PLS wITH n - 4

square matrices in so-called reduced-form,in the first column:

four 4 x 4 Latinthe top row and

/t 2 3

\+12

Ttrere are j usti.e., 112,3,4 in

il Lb:fi llil

rnatrices

standard-

the top

ltz 34\ ltz 34\L.:[; '^;?1, Ld:lliiil

\+31 z) \aB zt)Permuting the last three rows of each of the four Latin squareLo,La,L.,L6 generates 6 Latin squares, and so there are 24 so-called

fo* 4 x 4 Latin squares in all. By standard-form we require thatrortr/ be 1,2,3,4 but not necessarily the first column.

282

-i.*' .-t;-"- -,6

these 24 standard-form Latin squa es with a subscript letter

superscript set of three numbers as they appear in column

For example we represent the PLS (Figure 4-l ,left panel) of

We identifyarbrc.,d and a

1, rows 2,314.

-typ. a234 by

il

23344112

FrcuRE 4.1: Albania 1999, scorr 2590 (Mickey Mouse) & 20o7, scott28l1 (Pink Panther).

we present examples of 10 different types of. 4 x 4 PLS from 8 differ-

ent countries: Albania, canada, Gambia, Guinea, Guinea-Bissau, Pitcairn

Islands, Portugal, and TYistan da Cunha issued from 1972-2009'

As of October 31, 2010, we have identified 172 PLS with precisely 100

issued. for the wwF 1988-2009 from 73 countries and 33 from Macau (31

of type b324) but o-nly of these 10 types (out of.24): The most popular (by

far) are:

283

o e234 (58): backwards circulant, block-Latin, symmetrici)(i

23344112

o . a432 (40): forwards circulant, block-Latin

/1234\o b324 (54): Sudoku form, magic sqrare [l i '^ ;l

The sheetlet in Fisure 4.1 (left panel) ,rr*" I-"J, Srrl*a234, which is

a one-step backwards circula,nt, and is 2 x 2 block-Latin in that the top left

2 x 2 corner of four stamps is the same as that in the bottom right corner,

and the top right corner of four stamps is the same as that in the bottom

left corner. This seerns to be the most popular type of PLS-we have found

58 examples as of october 31, 20L0. The starnps were issued by Albania in1999 and feature Mickey Mouse.

The sheetlet in Figure 4.1 (right panel) forms a PLS of type a432, which

is a one-step forwards circulant, which is also popular-we have identified

40 examples as of October 31, 2010. The sta.rnps were issued by Albania in1999 and feature the Pink Panther.

The matrix lf,eza1 and the array of 16 stamps in the Canada-1972 PLS

(Figure 4.2, left panel) are in Sud,olru fonrr, i* that the four stamps each

appear once in the top left, top right, bottom left, and bottom tight 2 x2corners, and so the full Latin square is a solution to a 4 x 4 mi.ni,-Sudoku

ptnzle;Yates [39] says that such Latin squares have "balanced corners". For

more about Sudoku see, e.g., 11, 3, 4, 34, 35, 3Bl.

The PLS in Figure 4.2 (righi pa,nel) is essentially the only one3 we have

found of type a342, which is also in Sudoku form, like the PLS of. type a324

from Canada (Figure 4.2,left panel). Depicted is the subantarctic fur seal

(Arctocephalus tropicalrs). The stamps were issued in 2004 by Tbistan da

Cunha, a group of islands in the south Atlantic.The sheetlet shown in Figure 4.3 forms a PLS of type b324, which is

in Sudoku form and forms a magic squaxe; this type is very popular. This

sheetlet was issued by Canada in 2005 and featurm 4 Canadian bridges as

described in detail on the sheetlet-verso (Figure 4.4): The Jacques CartierBridge across the St. Lawrence River, the Angus L. Macdonald Bridge across

Halifax Harbour, thetsanso Causeway connecting the Nova Scotia mainland

rvith Cape Breton Island and the Swinging Bridge in Souris, Manitoba.

3W. have found 3 more PLS of type r,342 embedded in larger sheets of stamps [16].

23124134

284

(right) Tlistan da Cunha 2004, Scott 747-754'

k

Ftcuns 4.2: Canada 1972, Scott 582-585;

Frcunn 4.3: Canada 2005, Scott 2100*2103'

Frcune 4.4: canada 2005, scott 2100-2103 (verso).

285

-:+ T.:*

(_)

"*

FrcuRe 4.b: Guinea-Bissau 2002, Grcth4 304; {right) Portugal 1997, Scott2153-2156-

The PLS in Figure 4.5 (teft pa.nel), which is the only one of type b342 thatwe have identified, is also in Sudoku form and also forms a magic square'

this time with both diagonals "Latin" in that each starnp appears once in

the main forwards and backwards diagonals. Depicted is the African buffalo

(Synceras caffer), a large African cloven-hoofed mammal.

The sheetlet in F igure 4.5 (Ieft panel) was issued by Guinea-Biss att (Gui,n€-

Bi.ssau), in western Africa; formerly Portuguese Guinea, upon independence,

it added the name of its capital, Bissau, to the country's name in or-

der to avoid confusion between itself and the nearby Republic of Guinea

{Rd.pubti.que d,e Guin6,e), which was formerly French Guinea (and which is-

sued the PLS of type c423 shown in Figure 4.6).

This PLS of type b423 (Figure 4.5, right panel) also forms a magic square

with both diagonals "Latin" in that each stamp appeaxs once in the main

forwards and backwards diagonals; the only other such type is b342 (Figure

4.5, left panel). Issued by Portugal in 1997, the stamps shown in Figure 4.5

(right panel) depict the Pyrenean desman (Galemys pyrenaicus), a small

semi-aquatic mammal that lives in the Iberian peninsula; its body is like

that of a mrrskrat, its nose like that of a hedgehog and its feet like that of

a duck-billed platypus. The desman lses its snout as a periscope to breathe

above the water's surface.

These two PLS (Figure 4.5, the two pa^nels), respectively of type 6342

andb423, are orthogonal to each other in that the pair defines a philatelic

4li"t"d on the wwF Grorh AG website {13] but apparently not listed by scott {261.

286

11 16\

ln'rl4 r)2r"'r16

,,Graeco-Latin square" from which, using "Euler's algorithm"

well [9, p. 212), we find

Ig 41l+ 3 .Z\2 14

which is a fully-magic matrix in "classic" form in that the num

each appear once.

D6nes & Keed-

6

T2

3

13

bers 1,

FIGURE 4.6: Guinea 2OOO, Groth275,PLS type c423'

The sheetlet in Figure 4.6 was issued by the Republic of Guinea (R€.publique

d,e Gui.n6.e). Depicted is the Guinea-baboon {Papi,o papio) in row 1, columns

1 and 2, and the collared mangabey {cercocebus torquatus) in row 1, columns

3 and 4. The Guinea-baboon is a baboon from the otd world monkey fam-

il1.' ri.ith reddish brown hair and a hairless, dark-violet or black face' The

collared mangabey is a long-tailed and long-legged, relatively Iarge mon-

ke1' *'ith gsr*xpicuous white eyelids and short, backwardly-directed hairs on

the crori-n. This PLS is of type c423 and the only such PLS that we have

identifieci.The sheet,ei sh,r+-n in Figure 4.7 (lett panel) was issued by the Repub-

lic of The Garir:a. cc::rmonl1' known as Gambia, the smallest country on

the African cc,i,t-.e:ti- inainland. It is bordered to the north' east, and

south by Senegaj. .t = :--. a small coast on the Atlantic Ocean in the west'

Depicted is tire B-a:.'C:-'.'":,=i Crane (Balearica paaon'i'na), a bird in the

family Gntt'dae-. Ti--= .;'':-=s :-:-i lhe closely related Grey Crowned Crane

(Baleari,ca reguior:-:'.. -,'.':--,:- :::-'::: n-etter habitats for foraging, are the

only cranes that ca:' :--:-r: -:. ::::: T::= PLS is of type b432, which is in

Sudoku form and is ;- :.--- -- -

t

=*re_iEa

e

FIGURE 4.7: Gambia 2996, Scott 3014ad; (right) Pitcairn Islands 2007, Scott6lTad.

Our 10th and last example of a PLS (Figure 4.7, right panel) is of type

d324 and the only PLS of this type that we have found; we caII this typecriss-cross since the same stamp appears in each cell in the main forwards

diagonal, and the same stamp appea,rs in each cell in the main backwards

diagonal. The stamps were issued by the Pitcairn Islands, well known as thehome of the descendants of the Bounty mutineers and the accompanying

Tahitians. In the top row, left to right, are the black noddy (Anous m'i'nu-

lus), the brown noddy (Anous stoli.dus), the black-grey ternlet {Procelsternacerulea) and the sooty tern (Sterna fuscata).

4.1. 4 x 4 Rainbow Latin squares. The sheetlet of 16 stamps from Pak-

istan (Figure 4.8) is almost a PLS---even though 4 different water birds each

appear 4 times, the 4 copies of the stamp for each bird are not all identi-

cal. We may think of this as a ore-step backwards circulant PLS with a

superimposed "rainbow-square" with each row defining (the 4 parts of) a

rainbow. We will call this a rainbow-Pls (R-PLS), more precisely a lotr-rainbow-Pls (RR-PLS). Depicted are 4 kinds of water birds: (a) gadwall,

(b) common shelduck, (c) mallard, and (d) greylag goose. This is the only

RR-RLS that we have found.We have, however, found a column-rainbow-PlS (CR-PLS). Issued by

Nepal in 2005, this PLS (Figure 4.9) is of the very rare fully-magic PLS-

type b342 with the 4 stamps in each column marked by a different colour.

Depicted are 4 kinds of mammals: (a) gangetic doiphin, (b) Indian pangol.in,

(c) Asiatic white elephant, and (d) clouded leopard.

288

FtcuRe 4.8: Pakistan 1992, Scott 790ad.

Flcune 4.9: Nepal 2005, Scott 762ad.

289

5. PLS wITH n:5

We now look at PLS with n: Slorwhich we have identified just 17 examples

(including 8 embedded in two panes each with 100 stamps) with just 4

different patterns. We have identified 5 x 5 PLS from 6 "countries" (stamp-

issuing authorities): canada (8), Pakistan (1), Swaziland (1), Tbanskei (5),

T\rrkey (1), and the USA (1). These 17 PLS are all of the circulant type,

either one- or two-step backwards (c18, c2B) or forwards (c1F, c2F).

A 5 x 5 Latin square is said to be in stand,ard,-forrn when the first row is

1,2,3,4,5 and in red,uced,-forrn when it is in standard form and in addition

the first column is 1,2,3,4,5. Ftom the well-known stati,sti,cal Tables by

Fisher and Yates [12, Table XV, p. 86], we find that there are 56 reduced-

form 5 x 5 Latin squaxes (in contrast to just 4 reduced-form 4 x 4 Latin

squares and just 1 reduced-form 3 x 3 Latin square) yielding 24x56 :1344possible distinct types of 5 x 5 PLS in standard-form. Of these 1344 types

we have identified PLS of just these 4 types: C1B' CLF, CZB, C2F:

C1B -

CZR -

2 3 4

3 4 5

4 5151212 3

2 3 4

4 5112 3

3 4 5

512

C1F -

C2F _

2 3 4

12 3

5124 513 4 5

2 3 4

5123 4 5

12 3

4 51

1)

;)

5.1. One-step circulants; C1B, C1F. We have identified 5 PLS of the

one-step backwards circulant type C1B: 1 from Swaziland and 4 from Canada

and 10 of the one-step forwards circulant type c1F: 5 from Tlanskei, L from

T\rrkey and 4 frdm canada. The s PLS from canada are embedded in two

panes each of size 10 x 10.

A 5 x 5 PLS of type c1B was issued by swaziland in 1982 for "wildlifeConseryation", the flrst of a series of 4 sets of five stamps, see Figure 5.1-

This first set was one of the earliest sets of stamps issued for the wwF.Depicted in Figure 5.1 is PeI's fishing-owl (Scotopelia peli,), a large species

of owl which feeds nocturnally on fish snatched from the surface of lakes

and rivers. Printed on the stamps are the words Si,khoua setinhlanti,, which

is fishing-owl in siswati, a Bantu language spoken in swaziland and south

Africa. Depicted are (top row left to right) male, female, pair, nest and egg,

29A

adult (with youngster). The Kingdom of Swaziland is a

in southern Africa, bordered by South Africa on three

east, where it borders Mozambique.

landlocked countrysides except to the

FIcuRE 5.1. Swaziland 7982, Scott 405ae, PLS type C1B. (Image courtesy Marc Steurs {28].)

Pel's fishing owl is named after Hendrik Severinus Pel (1818-1876), who

was a member of the technical staff of the Rijksmuseum van Natuurlijke His-

torie in Leiden from 1832-1840 and then worked for the Dutch government

of the Gold Coast with the understanding that in his spare time he would

collect natural history specimens for the Leiden Museum. In the mid-19thcentury, The Netherlands occupied six trade posts in what is now Ghana.

291

These stamps from Swaziland (Figure 5.1) seem to be quite rare. The

2010 Scott Stand,ard Postage Stamp Catalogue iists a single stamp, unused,

at US$16.00 and a strip of five stamps, unused, at US$125.00, while on the

Steurs WWF website the full PLS is for sale at US$1609 on October 31, 2010

(US$1655.20 on January 29,2010, up from US$1483.10 on June 12, 2008).

Delcampe [8] has a strip of 5 stamps (row 2) for sale at 62 Euros (US$86.46)

on October 31, 2010, while the Groth WWF website lists these stamps as a

"forerunner" (Groth #F40) of those issued for the WWF and lists a stripof 5 stamps for sale 150 Swiss francs (US$152.28 but not in stock).

FIcURE 5.2. T\rrkey 196O, Scott 1488-1492, PLS type C1F.

A 5 x 5 PLS of type CF1 was issued in 1960 by T\rrkey (shown in Figure5.2), the earliest 5 x 5 PLS that we have identified (the earliest 2 x 2 PLSthat we have found was issued by Canada in 1960). This PLS was issued

for the 1960 Summer Olympics held in Rome, Italy, August 25-September

11, 1960. Rome had been awarded the organization of the 1908 Summer

Olympics, but after the 1906 eruption of Mount Vesuvius, was forced todecline and pass the honors to London. At the 1960 Olympics, Turkey won

7 gold medals and 2 silver-all 9 for wrestling. Depicted on the stamps are

(top row, left to right): hurdling, soccer, steeplechase, basketball, wrestling.

292

-S: .,"{a

5.2. 5x5-embedded PLS.

Frcunn 5.3. Canada L97A, Scott 519-523.

Four 5 x 5 PLS are embedded in each of two panes of 100 stamps, one pane

with 5-cent stamps (Figure 5.3), the other with 6-cent stamps. These panes

of stamps were issued by canada for christmas 1970 featuring children's

designs. The pattern in each of the two panes is the same:

/o0001\

(!. ii):(I)or, F) , F:[3 3 ? ll)\FA FAF/ \F/ --'- - t '

\l I 3 o,

olHere the 5 x 5 matrix A denotes the one-step backwards circulant matrixC1B, the matrix I the 5 x 5 identity matrix, and F the 5 x 5 flip matrix' The

PLS defined by AF (top right corner) and FA (bottom left corner) are one-

step forwards circulants of type C1F and FAF (bottom right corner) is a

one-step backwards circulant of type C1B like the PLS defined by A (top left

corner), but with the stamps arranged differently. There are, therefore, eight

distinct PLS in all, with two one-step forwards circulants and two one-step

backwards circulants ln each of the two panes. An interesting feature of these

stamps is that just one of the five stamps appears four times contiguously

in a block of four and so only in the centre of the full pane (Figure 5.3).

293

FIGURE 5.4. Pakistan 2OO4, Scott 1045ae; PLS type C2B.

5.3. Two-step circulants: C2B, C2F. The only 5 x 5 PLS of the two-step circulant types C2B and C2F that we have identified are from Pakistan(Figure 5.4) and the USA (Figure 5.5). These PLS have several interestingproperties. In particular they are both magic-Latin in that each of the 5stamps appears once in the two main diagonals. Moreover, {23]: "Of Latinsquares used for crop experiments with a single set of treatments, the earliestexamples (published in L924 [36]) are 5 x 5 squares of the types {C2B andC2Fl known as Knut Vik or "knight's move" designs; these are squares whereall cells containing any one of the treatments can be visited by a successionof knight's moYes (as in chess) and where no two diagonally adjacent cellshave the same treatment."

As noted by Fisher in The Desi,gn of Erperiments ftl, p. 7B]: "In thisarrangement the areas bearing each treatment are nicely distributed overthe experimental area, so as to exclude all probability that the more impor-tant components of heterogeneity should influence the comparison betweentreatments." In 1931 Tedin [33] and in 1951 Nissen [21] observed that thereare just two possible arrangements of the 5 x 5 Knut Vik design in standard-form: C2B and C2F. These designs are pandiagonal, in that all diagonals(the two main diagonals and all broken diagonals) contain each of the fivetreatments precisely once.

294

FIcURE 5.5. USA 1995, Scott 3019v; PLS type C2F.

A PLS of type C2B was issued by Pakistan (Figure 5.4) for the Na-

tional Philatelic Exhibition in Lahore on Universal Postal Union Day (Oc-

tober 9, 2004). This PLS depicts five kinds of tropical fish (top row, leftto right): striped gourami (Coli,sa fasciata), black widow tetra (Gyrnnoco-

rymbus ternetzi), yellow dwarf cichlid (Apistogramma borellii), tiger barb(C apo eta tetrazona), neon tetra { P arachei,rodon i,nn esi,) .

A PLS of type C2F was issued by the USA (Figure 5.5) depicting flveantique automobiles from the late 19th and (very) early 20th century. Inthe top row (left to right): 1893 Duryea, 1898 Columbia, 1901 White, 1894

Haynes, 1899 Winton.

6. AoxNowLEDGEMENTS

We are most grateful to Marc Steurs [28] for the image of the (very rare) PLS

in Figure 5.1; all other PLS images are from the personal collection [31] of the

second author. Information about philatelic Latin squares may be found in theopen-access websites {2, 8, 10, 13, 14, 79,20,281, as well as in the Scotf stamp

catalogues [26]. Much of our biographical and geographical information is based onthat in Wi.kiped,ia [37]. Many thanks to S. Ejaz Ahmed, Nicolas Ammerlaan, Oskar

Maria Baksalary, Christian Boyer, Ka Lok Chu, Jeffrey J. Hunter, Ian D. Kim-merly, Alexander Kovaiec, Sid Kroker, Sunil Kukreja, Michael D. Larsen, FliedrichPukelsheim, Simo Puntanen, Daniel J. H. Rosenthal, Evelyn Matheson Styan, Gcitz

tenkler, and Kimmo Vehkalahti for their help. Research was supported, in part,by the Natural Sciences and Engineering Research Council of Canada.

295

RerpReNces

11] R. A. Bailey, Peter J. Cameron & Robert Connelly (2008). Sudoku, gerechte de-

signs, resolutions, affine space, spreads, reguli, and Hamming codes. ?ft.e AmericanMathernati.cal Monthly, 115 (5), 38}4;04. L2

[2] BiStamp.com. Stamp professionals specializing in postage stamps of Russia and otherex-USSR states and worldwide topicals: online open-access website at Kyiv, Ukraine.6,23

[3] Christian Boyer (2007). Sudoku's F]ench ancestors. The Mathematical Intelligencer,29 (1),37-44. [Revised and enhanced version (in English) of the original F]enchversion: "Les anc6tres frangais du sudoku" ia Pour La Science, no. 344, pp. 8-11 &89, June 2006.1 12

[4]

i5l

Christian Boyer (2007). Sudoku's Flench ancestors: solutions to problems. The Math-ematical Intelligencer,29 (2), 5H3. 12Ka Lok Chu, Simo Puntanen & George P. H. Styan (2009). Some comments onphilatelic Latin squares from Pakistan. Pakistan Journal of Statistics,2S (4),427-471: online open-access. 1

[6] Charles J. Colbourn, Jeffrey H. Dinitz & Ian M. Wanless (2OO7). Latin squares. InHand,book of Combi,natori,al Designs Charles J. Colbourn & Jeffrey H. Dinitz, eds.),Chapman & Hall/Taylor & Flancis, Boca Raton, pp. 135-152 ($ III, ch. 1). 1

[7] [Frangois] Crett6 de Palluel (1788). Sur Ies avantages & I'6conomie que procurentles racines employ6es i, l'engrais des moutons e l'6table. Mdrnoire-s d,'agri.culture,d.'6eonomi,e rurale et d,amestique, Tlimestre d'6t6 1788 ("Lu le 31 juillet 1788"),pp. 17-23 (& 2 foldout-pages of tables facing p. 23).2

{8] S6bastien Delcampe: delcampe.com. The international marketplace for [stamp] col-lectors: online open-access website in Soignies, Belgium. 20, 23

[9] J. D6nes & A. D. Keedwell (1,974). Lati.n Squares and, thei.r Applications. AcademicPress, New York and Akad6miai Kiad6, Budapest. 15

{10] eBay.com. Open-access auction and shopping website: online for stamps. 23

[11] R. A. Fisher (1990). Statisti.cal Methods, E erimental Design, and. Scientific Infer-erlce, a re.issue of three books in one of: (book L) Statisticel Me.thods for R.esearch

Workers,(book 2) The Design of Erperiments, and (book 3) Statistical Method.s and.

Scientifi.c Interence, edited by J. H. Bennett. Oxford University Press. [Reprintedwith corrections 1995.] 22

[12] Sir Ronald A. Fisher & Frank Yates (1963). Stati.stical Tables for Biologi.cd, Agri-cultural and, Medical Research, Sixth Edition, Revised and Enlarged. Oliver & Boyd,Edinburgh; online open-access: R. A. Fisher Digital Archive, Adeiaide Research andScholarship. 18

[13] Groth AG. WWF Conservation Stamp Collection: online open-access website inUnterd.geri (canton Zug), Switzerland. 5, 14, 23

[14] Library and Archives Canada (2008). Canadian Postal Archives Database: onlineopen-access search by keyword or by year. 23

{15] [Ramon Llull (1232-1316)j (2007). Rairnund.i Lulli Opera Lati,na 27: Ars Dernon-strzti,ua, edited by (cura e studi,o) Josep Enric Rubio Albarracin. Text in Latin witirintroductions in Spanish and foreword in German. Corpus Christianorum ContinuatioMediaevalis 213, Raimundi Lulli opera Latina 32. Brepols, T\rrnhout. 2

[16] Peter D. Loly & George P. H. Styan (2009). Philatelic Latin squares. Report 2009-01,

Department of Mathematics and Statistics, McGill Universitg Montr6al. 1, 12

[17] Peter D. Loly & George P. H. Styan (2010). Comments on 4 x 4 philatelic Latinsquares. Chance,23 {l),5742 (Winter 2010). 1

[18] Peter D. Loly & George P. H. Styan (2010). Comments on 5 x 5 philatelic Latinsquares. Chance,23 (2),5842 (Spring 2010). 1

[19] Marlen Stamp & Coins Ltd: online open-access website in Great Neck, New York. 23

[20] B. Mclean Stamps for Collectors: P.O. Box 323, Ellon, Aberdeenshire AB41 7YA,Scotland UK. online open-access website. 3, 23

296

[21] @ivind Nissen (1951). The use of systematic 5 x 5 squares. Biometrics, T (2), L67-170;online at JSTOR. 22

[22] Lewis P. Orans (2004). Reverend Professor Baden Powell, Savilian Professor of Ge-ometry at Oxford: online open-arcess. 2

[23] D. A. Preece (i990). R. A. Fisher and experimental dasign: a review. Biometrics, 46

(4), 925-935: online at JSTOR. 2,22124) Teny Ritter (2003). Latin squares: a literature survey. FUesearch comments ftom Ci,-

phers By Ritter: online open-access, last updated: 2003-L2-21 (from 1998-05-29). 1

[25] William L. Schaaf (1978). Mathematics and Science: An Ad,uenture 'i,n Postage

Stamps. National Council of Teachers of Mathematics, Reston, Virginia. 1

[26) Scott Stand,ard Postage Stamp Catalogue (James E. Kloetzel, ed.), currently published

annually in several volumes by Scott Publishing, Sidney, Ohio. 2, 4, L4,23

[27] David Curtis Skaggs (1978). Postage stamps as icons. ln lcons of America (Rav B.

Browne & Marshall Fishwick, eds.), Popular Press, Bowling Green, Ohio, pp. 198-208.1

{28] Marc Steurs. The most complete site for your WWF stamp collection: online open-access website at Rijmenam (near Antwerp), Belgium. 5, 19, 23

[29] George P. H. Styan {2007). A philatelic introduction to magic squares and Latinsquaxes for Euler's 300th birthyear.In Proeed'i,ngs o! the Canad,ian Society for Historyanil Phi.losophy of Mathematics, vol.20, pp. 306-319. 1

[30] George P. H. Styan (2009). An illustrated philatelic introduction to 4x4 Latin squares

in Europe; 1283-1788 (with some comments about Ie fauteui,l Dagobert and aboutphilatelic Latin squares in Europe: L984-2007). Talk presented at the 18th Interna-tional Workshop on Matrices and Statistics: fWMS'O9, Smolenice Castle, Smolenice,'Slovakia, 26 June 2009, overheads pdf 6le, 28 pp. 1

[31] George P. H. Styan (2010). Personal collection of philatelic items. Verdun (Qu6bec),Canada, October 31, 2010. 23

[32] George P. H. Styan, Christian Boyer & Ka Lok Chu (2009). Some comments on Latinsquares and on Graeco-Latin squares, illustrated with postage stamps and old playingcards. Statistiutl Papers,50 (4), 917-941: online at Springerlink.l

[33] Olof Tedin (1931). The influence of systematic plot arrangement upon the estimateof error in field experiments. The Jourt'tal of Agricultural Science,27, l9L-208. 22

[34] Peter Ullrich (1999). An Eulerian square before Euler and an experimental designbefore R. A. Fisher: on the early history of Latin squares. Chance, 1,2 (L) , 22-26. 72

[35] Peter Ullrich (2002). Ofrcers, playing cards, and sheep: on the history of Euleriansquares and of the design of experiments. Metrilea,56 (3), 189-204: online at Springer-Link. 12

[36] Knut Vik (1924). Bedtmmelse av feilene pi forsOksfelter med og uten mS.lestokk.Meldinger fra Norges Landbraksh.ogsleole, 4, L29-181.22

[3fl Wikiped.ia, The Flee Encyclopedia. Wikimedia Foundation: online open-access web

archive. L,2, 4,23[38] Robin Wilson (2006). How to Solae Sud.oku: A Step-by-Step Gui.de. Sterling, New

York. 12

I39] F. Yates (1939). The comparative advantages ofsystematic and randomized arrang+ments in the design of agricultural and biological experiments. Biometrika, S0 (3/4),44A-466: online at JSTOR. 12

PBreR D. Lolv: DepeRrrasNr oF PHysrcs AND AsrRoNoMy, THE Uxtveastrv orMaulroea, WtlNrroc, Iv{eNrroae, CnNepa RgT 2N2

E-mail add.ress: 1oly@cc . umantioba. ca

GsoRce P. H. SrveN: DspeRTru,rpNT oF MATHEMATICS AND STATISTICS, McGILLUNIveRsrry, 805 ouesr, RUE SHERBRooKE, MoNTR6AL (QuEBEc), CeNloe H3A 2K6

E-mail add,ress : styaa@nath. ncgi11. ca

",

297

'i!

PROCEEDINGS OF THE

Canadian Society for Historyand Philosophy of Mathematics

Soci6t6 Canadienne d'Histoireet de Philosophie des Math6matiques

$CHFl\ACSHPIVI

Volume 23

T h i r ty -fift h A n n u a I M e e t i n g/-R d u n i o n A n n u e I I e

Concordia University/Universit6 ConcordiaMontr6al, Qu6bec, Canada

May 29-31,2010

? €q'r'a-_:5_ !:+ r\ i:-.!

Preface

The CSHPM/SCHPM held its 2010 Annual Meeting at Concordia University, Montrdal,

Qu6bec, on May 29-31, 2010. These Proceedings include the majority of the paperspresented at the Annual Meeting.

I would like to thank the Executive Council of the Society for giving me the chance to be

the editor again, and the contributors to the present volume for their cooperation andkindness.

Antonella CupillariPenn State Erie-The Behrend College

February L9,2011

ffi -f;r{-a:;p.-<+$'_j'

Table of Contents

Preface

rancine F. AbelesThe Early History of Quasideterminants in the Nineteenth Century . . . . . 1

Amy Ackerberg-HastingsWhat Is a Great Book? A Case Study of Legendre's El,iments de Gdometrie (1794)

and Playfair's Elements of Geometry (1795) . . .. .7

Christopher BaltusCentral Collineations in 1673:The Nouvelle Mdthode andthe Plani-Coniques of Philippe De La Hire . . . . .l4

Janet Heine BarnettMathematics is a Plural Noun: The Singular Case of Oliver Byrne . . . . 26

David R. BellhouseThe Mathematics Curuiculum in the English Dissenting Academies . . . . 47

Robert E. BradleyCauchy'sAlgebraicAnalysis ....57

Edward L. CohenTheRomanCalendars ....72

,/ l.4ichael Cuffaro\/ Wittgenstein on Prior Probabilities .. . . . 85

Antonella CupillariElisha Scott Loomis and Proof Techniques . . . . . 99

Lawrence D'AntonioDid Quadratic Forms SpringFull-Blownoutofthe HeadofGauss? .. ...... 107

Scott DixonConcreteModalStructuralism.... ....119

Emerson P. DoyleRamsey's LittleArgument... ...132

Thomas DruckerMathematicsasaliberalArt... ......155

Roger Goda.rd<LA GEOMETNE DES FORMEST The Paths to Computational Geometry . . . 159

A Ll

Hardy GrantMathematics and the Liberal Arts: The Beginnings . .175

AIex KooTheMiddleRoad:AResponsetoColyvan .....1g3

Gregory LaversOn the Quinean-analyticity of mathematical Propositions . . . . lg2

Menolly LysneWhy do we not remember Laplace's Work on Singular Solutions? . . . . 205

David OrensteinThe Real and ldeal in High School History and Philosophy of Mathematics . . 227

Andrew B. PerryNicolas Pike's New and Complete System of Arithmetic . . . . . . .233

Bruce J. PetrieBradley's reference Leonhard Euler's (Jse and (Jnderstanding ofMathematicalTranscendence ...241

Josipa G. PetrunicColonial Mathematics:The spread of Metropole Knowledge In The Nineteenth century . . . . .z5l

Dirk Schlimm and Katherine SkosnikSymbolsfor Nothing: Dffirent Symbolic Roles of Zero andTheir Gradual Emergence in Mesopotamia. . . .257

Charlotte K. SimmonsYesudas Ramchundra: De Morgan's Ramanujan? .... .......267

James T. SmithDefinitions and Nondefinability in Geometry:Legacies of Mario Pieri and Alfred Tarski (abstract). . . .Z7Z

fieorge P. H. Styan and Peter D. Loly\ / Philatelic Latin Squares . . . .\./ nquures ..---.-273

James J. TattersallThe Prouhet-Tarry-Escott Problem . ... 2gg

Robert ThomasWhy a Mathematician might be (a bit) interested in Theodosios's "spherics . . 305

/Maryam Vulis

1/ AndreiAndreyevichMarkov ....310

I

October 31, 2010

PHILATELIC LATIN SQUARES

PETER D. LOLY AND GEORGE P. H. STYAN

Abstract. Postage stamps are occasionally issued in sheetlets (smallsheets, miniature sheets, souvenir sheets) of n different stamps printedin an n × n array with n copies of each of the n stamps. We focus onsheetlets that form a philatelic Latin square (PLS): each of the n stampsappears exactly once in each row and exactly once in each column. Wegive examples for n = 2, 3, 4, 5 building on results in [5, 16, 17, 18, 30];see also [29, 32].

Resume. Des timbres sont parfois emis en planches (petites, miniatures,souvenirs) de n differents timbres imprimes dans un tableau n× n avecn exemplaires de chacun des n timbres. Nous nous concentrons sur lesplanches qui forment un carre latin philatelique (PLS en anglais) : cha-cun des n timbres apparat exactement une fois dans chaque ligne etexactement une fois dans chaque colonne. Nous donnons des exemplespour n = 2; 3; 4; 5 en nous basant sur les resultats de [5, 16, 17, 18, 30];voir aussi [29, 32].

1. Introduction and mise-en-scene

Postage stamps are occasionally issued in sheetlets (small sheets, miniature

sheets, souvenir sheets) of n different stamps printed in an n×n array with

n copies of each of the n stamps. Here we focus on sheetlets where each of

the n stamps appears exactly once in each row and exactly once in each

column. We call such a sheetlet a philatelic Latin square (PLS) and have

found PLS for n = 2, 3, 4, 5; we would be pleased to learn of any PLS with

n ≥ 6. For more about Latin squares see, e.g., [6, 24].

The mathematics educator William Leonard Schaaf (1898–1992) in his

Mathematics and Science: An Adventure in Postage Stamps [25, p. xiii] ob-

served that “The postage stamps of the world are, in effect, a mirror of civi-

lization”, while Skaggs [27] refers to postage stamps as “Icons of communica-

tion”. Morever, “Philately is the study of stamps and is distinct from stamp

collecting—it is possible to be a philatelist without owning any stamps!” [37]

and at this 2010 Annual Meeting of the CSHPM: “Mathematical philately

is a liberal art!”1

2000 Mathematics Subject Classification. 01A45, 01A50, 01A55, 01A60, 05-03, 05A99,05B15.

Key words and phrases. almost-PLS, balanced corners, block-Latin square, block-PLS,embedded-PLS, Europa CEPT stamps series; European Conference of Postal and Telecom-munications Administrations (CEPT), experimental design, knight’s move design, KnutVik design, magic-Latin, miniature sheet, philatelic Latin square (PLS), postage stamps,rainbow-PLS, souvenir sheet, sheetlet, Sudoku form, topical philately, World Wide Fundfor Nature (WWF), World Wildlife Fund.

1This paper is based on the talk presented (by George P. H. Styan) in the SpecialSession on Mathematics and the Liberal Arts at the Annual Meeting of the CanadianSociety for the History and Philosophy of Mathematics, Concordia University, Montreal(Quebec), Canada, May 31, 2010, and, in part, on [5, 16, 17, 18, 30].

2 PETER D. LOLY AND GEORGE P. H. STYAN

We were surprised to find that Latin square designs have been used in

printing postage stamps, with the first such use apparently being in

• 1949 USSR 2 × 2: Scott 1359–1360 (Figure 2.1, left panel),

• 1960 Turkey 5 × 5: Scott 1488–1492 (Figure 5.2),

• 1972 Canada 4 × 4: Scott 582–585 (Figure 4.2, left panel),

• 1976 Isle of Man 3 × 3 : Scott 86–88 (Figure 3.2, left panel).

Throughout this paper Scott catalog numbers are as found in the Scott Stan-

dard Postage Stamp Catalogue [26].

Latin square designs of size 4× 4 have been used in Europe at least since

the 13th century, when the Majorcan writer and philosopher Ramon Llull

(1232–1316) published the “First elemental figure” in his Ars Demonstrativa

[15] in 1283. As described by Preece [23], probably the earliest use of a Latin

square in an experimental design was by the agronomist Francois Crette de

Palluel (1741–1798), who in 1788 published a study [7] of an experiment

involving the fattening of 16 sheep in France, four of each of four different

breeds. The advantage of using a Latin-square design in this experiment

was that only 16 sheep were needed rather than 64 for a completely cross-

classified design. The purpose here was to show that one might just as well

feed sheep on root vegetables during the winter; this was much cheaper, and

easier, than the normal diet of corn and hay.

2. PLS with n = 2

There are many 2 × 2 PLS but the oldest we have identified is from the

USSR (Figure 2.1, left panel) issued in 1949 in celebration of the 150th

birth anniversary of Alexander Sergeyevich Pushkin (1799–1837), who is

considered to be the greatest Russian poet [37]. The stamp in the top-left

corner shows Pushkin in 1822 and the stamp in the top-right corner is based

on a portrait by Orest Adamovich Kiprensky (1782–1836) painted in 1827.

This is the oldest PLS (of any size) that we have identified to date.

The 2×2 PLS (Figure 2.1, right panel) was issued by Romania in celebra-

tion of the centenary of the Scouting Movement. The stamp on the top-left

of the PLS shows its founder, Robert Stephenson Smyth Baden-Powell, 1st

Baron Baden-Powell (1857–1941), whose father, Reverend Professor Baden

Powell (1796–1860), was Savilian Professor of Geometry at Oxford [22]. This

PLS was issued in the “Europa” series. Europa stamps have been issued an-

nually since 1956; from 1959–1992 the Europa series stamps represented the

European Conference of Postal and Telecommunications Administrations

(CEPT2). From 1993 to date the stamps represented PostEurop [sic], the

2The acronym CEPT comes from the French: “Conference europeenne des administra-tions des postes et telecommunications”.

PHILATELIC LATIN SQUARES 3

Association of European Public Postal Operators, one of the constituent

unions of the Universal Postal Union. There are many 2 × 2 PLS issued in

the Europa series and we recommend the McLean website [20].

Depicted on the stamps are Mustafa Kemal Atatürk (1881–1938), founder of the Re-public of Turkey as well as its first President, and Quaid-e-Azam Muhammad Ali Jinnah(1876–1948), who is generally regarded as the founder of Pakistan [80]. One stamp depictsonly Atatürk. These stamps were issued in 5× 4 sheetlets of 20 stamps (10 of each of 2different stamps) in a chessboard (or checkerboard) pattern (Figure 1.2). There are twodistinct 2×2 philatelic Latin squares (PLS) embedded in this sheetlet. The first is, e.g., inthe top-left 2× 2 corner of Figure 1.2 as shown in Figure 1.1. The other 2× 2 PLS, withthe two stamps switched is, e.g., in the bottom-right corner of Figure 1.2. Each 2×2 PLSis repeated 6 times (4 times disjointly) in the full sheetlet.

FIGURE 1.3: USSR 1949, Scott 1359–1360.

Many countries have issued postage stamps in sheets or sheetlets with a chessboardpattern with various sizes. There are several sheetlets also which depict just 2 stampseach twice and form a 2×2 PLS. An example is the sheetlet (Figure 1.3) issued by USSRin 1949 in celebration of the 150th birth anniversary of Alexander Sergeyevich Pushkin(1799–1837), who is considered to be the greatest Russian poet [80]. The stamp in the

5

Figure 2.1: USSR 1949, Scott 1359–1360; (right) Romania 2007, Scott 4941–4942.

Figure 2.2: Isle of Man 1981, Scott 198–199; (right) Great Britain 2005, Scott 2279ab.


A 2× 2 PLS (Figure 2.2, left panel) was issued by the Isle of Man in cele-

bration of the marriage of The Prince Charles, Prince of Wales (b. 1948) to

Lady Diana Spencer (1961–1997) on 29 July 1981 and a 2 × 2 PLS (Figure

2.2, right panel) was also issued by Great Britain in celebration of his mar-

riage to Camilla Parker Bowles (b. 1947) on 9 April 2005. Camilla, Duchess

of Cornwall is the current holder of the titles Princess of Wales and Duchess

of Cornwall and of Rothesay [37]. The marginal inscription (in English and

Welsh) states that the wedding took place on 8 April 2005, but it was de-

layed till the following day 9 April due to the Prince’s attendance at the

April 8 funeral of Pope John Paul II (1920–2005).

The Isle of Man is a self-governing British Crown Dependency, located in

the Irish Sea between the islands of Great Britain and Ireland, within the

British Isles. The head of state is Queen Elizabeth II, who holds the title of

Lord of Mann (sic). The island is not part of the United Kingdom, but its

foreign relations and defence are the responsibility of the UK Government.

Isle of Man stamps are indexed by Scott [26] under Great Britain.

2.1. 2×2 embedded-PLS.

Figure 2.3: Canada 2009, Scott 2326–2327.


There are many 2×2 PLS in sheetlets of 4 stamps but it seems there are even

more 2×2 PLS embedded in sheetlets of 4k stamps, arranged “checkerboard-

style”, with k ≥ 2. We will refer to such sheetlets as embedded-PLS. For

example, in the PLS issued by Canada in a series to “Preserve the Polar

Regions and Glaciers” (Figure 2.3), k = 4 and embedded is a 2 × 2 PLS

featuring a stamp with a polar bear (Ursus maritimus) on the main diagonal

and a 2× 2 PLS featuring a stamp with an Arctic Tern (Sterna paradisaea)

on the main diagonal. There are in all in this sheetlet 5 copies of the PLS

featuring the polar bear and 4 copies of the PLS featuring the arctic tern.

In Figure 2.4 we display two sheetlets issued for the World Wide Fund for

Nature (WWF), an international non-governmental organization working

on questions regarding the conservation, research and restoration of the

environment, formerly called the World Wildlife Fund, which remains its

official name in the United States and Canada. There are many PLS issued

for the WWF and we recommend the websites of Groth AG [13] and Steurs

[28].10/03/2008 05:05 PMWWF Stamp Collection :: Issues by Country :: Vietnam :: #274 2000 Sao La :: Vietnam Sao La Sheetlet of 2 sets / 8 stamps

Page 2 of 2http://www.wwfstamps.org.uk/club/product.php?productid=257503122&cat=3010&page=1

WWF Stamp Club - Join the Club, Save the Nature Copyright © 2007-2008 WWF Stamp Collection

Vietnam 2000 12/19/2008 06:46 PM# 1 in Mary. WWF

Page 1 of 2http://www.bistamp.com/inter/BiStamp.nsf/db7325a4ad874c65c2256e51006de94f/9e463731b4e2a0d9c2256cd90030d70d!OpenDocument

Mary # 1

WWF

$2.40

WANT TO PURCHASE THIS ITEM? Read our Ordering Information first.

Go to the Main page.

Note for visitors: you are now in a new window; close it to go back to the price-lists.

Figure 2.4: Viet Nam 2000, Scott 2966–2969; (right) Turkmenistan (Mary Velayat) 1998.

The Viet Nam sheetlet (Figure 2.4, left panel) features the Saola or Vu

Quang ox (Pseudoryx nghetinhensis), one of the world’s rarest mammals. It

is a forest-dwelling bovine found only in Viet Nam and in Laos, near the

Viet Nam–Laotian border. Rows 2 and 3 form a 2 × 2 PLS.

The other sheetlet (Figure 2.4, right panel) was issued by Turkmenistan

(Mary Velayat) 1998 and features “wild animals and birds”. Mary Velayat is

the southeastern region of Turkmenistan on the Turan Lowland, within the


limits of the Karakum desert. The top two rows, and the bottom two rows,

each form two 2 × 2 PLS which are “block-reflections” of each other, and

so embedded in this sheetlet are four 2 × 2 PLS. For stamps from countries

and certain regions that were part of the USSR we recommend BiStamp [2].

2.2. 2×2 block-PLS.

Figure 2.5: Macau 2008, Scott 1248ah. (Joint issue with Singapore 2008.)

Figure 2.6: Singapore 2008, Scott 1318ad–1319ad. (Joint issue with Macau 2008.)


There is a 2×2 block-PLS entitled “Local gastronomy” from Macau (Figure

2.5), issued jointly with Singapore (entitled “Local delights”), where each

block comprises 4 stamps, and so, 8 distinct stamps in all in a sheetlet of

16 stamps. The Latin square underlying a 2×2 block-PLS is a “block-Latin

square”. We do not know if the Singapore stamps (Figure 2.6) were also

issued as a 2×2 block-PLS and we welcome information about this.

Featured on the stamps are 4 Macau dishes: Carne de porco a Alentejana

(Pork and clams), Lombo de bacalhau braseado em lascas (Grilled salt cod

slices), Yangzhou fried rice, Crispy fried chicken and 4 Singapore dishes:

Roti prata (fried flour-based pancake), Hainanese chicken rice, Laksa (Spicy

noodle soup), Satay (Meat marinated in a spicy peanut sauce on a skewer).

Figure 2.7: Hong Kong 2003, Scott 1063–1068.


The only other 2 × 2 block-PLS we have found is from Hong Kong (Fig-

ure 2.7) issued for “World heritage in China”. Here each block comprises 3

stamps with the two blocks in the top “row-block” (of 6 stamps) repeated

in the bottom row-block; the sheetlet, therefore, includes 2 distinct 2 × 2

block-PLS (block-reflections of each other) one being the top 2 rows and

the other being the bottom 2 rows. The stamps feature (HK$1.40) Potala

Palace, ($1.80) Imperial Palace of the Hing and Qing Dynasties, ($2.40)

Mausoleum of the First Qin Emperor, ($2.50) Mount Huangshan, ($3.00)

Old Town of Lijiang, ($5.00) Jiuzhaigou Valley.

3. PLS with n = 3

There are just two types of 3×3 PLS: one-step backwards circulant (C1B) as

illustrated by the sheetlet depicting trees of the Holy Land issued by Israel

(Figure 3.1, left panel); depicted are the Greek strawberry tree (Arbutus

andrachne), Judas tree (Cercis siliquastrume), and Vallonea oak (Quercus

ithaburensis). This PLS (Figure 3.1, right panel) is the one-step forwards

circulant (C1F) issued by the “Grenadines of St. Vincent” in 1993 for Elvis

Presley (1935–1977); a PLS with the same design was also issued in 1993

by Dominica, Gambia, Maldives, Nevis, and St. Vincent (Scott 1767). As of

October 31, 2010, we have identified 45 (including 18 from Abkhazia) PLS

with n = 3: only 8 of type C1B and the other 37 of type C1F, including 6 for

Elvis Presley and 9 embedded (6 from Costa Rica and 3 from St. Vincent).

Figure 3.1: Israel 1981, Scott 798–800;(right) “Grenadines of St. Vincent” 1993, Scott 1347ac (listed under “St. Vincent Grenadines”).


Saint Vincent and the Grenadines is a nation which lies at the southern

end of the eastern border of the Caribbean Sea where it meets the Atlantic

Ocean. Its 389 km2 (150 mi2) territory consists of the main island of Saint

Vincent and the northern two-thirds of the Grenadines, which are a chain

of smaller islands stretching south from Saint Vincent Island to Grenada.

The oldest 3×3 PLS that we have identified is from the Isle of Man issued

in 1976. Issued for the Europa CEPT programme, this 3 × 3 PLS (Figure

3.2, left panel) is one of a pair featuring Manx ceramic art: (top row, left to

right) Barrosse beaker (Bronze Age), souvenir teapot (3-legged man, 19th

century), Laxey jug (1854). Another pair of 3 × 3 PLS were issued in 1978

featuring carved gravestones: (Figure 3.2, right panel, top row, left to right)

wheel-headed cross-slab, Celtic wheel-cross, Keeil Chiggyrt stone. All four

3×3 PLS issued by the Isle of Man in 1976/1978 are of the one-step forwards

circulant (C1F) type.

Figure 3.2: Isle of Man 1976, Scott 86–88 & (right) 1978, Scott 131–133.

3.1. 3×3 embedded-PLS. Three distinct 3 × 3 PLS are embedded in the

4 × 4 sheetlet (Figure 3.3) issued by “St. Vincent & The Grenadines” for

Sandy Koufax (b. Sanford Braun, 1935), the American left-handed former

baseball pitcher in US Major League Baseball who played his entire career

for the Brooklyn/Los Angeles Dodgers, from 1955–1966. All four 3× 3 PLS

are of the one-step forwards circulant (C1F) type with the PLS formed from

the first 3 rows and columns repeated in the PLS formed from the last 3 rows

and columns. We have also found six 3× 3 PLS embedded in sheetlets from

Costa Rica issued in 2005 (Scott 586ac, 596ac) depicting flora and fauna in

national parks (3) and indigenous fruits (3).


Figure 3.3: “St. Vincent & The Grenadines” 1996, Scott 2356ac (listed under “St. Vincent”).

4. PLS with n = 4

There are just four 4 × 4 Latin square matrices in so-called reduced-form,

i.e., 1, 2, 3, 4 in the top row and in the first column:

La =

1 2 3 42 3 4 13 4 1 24 1 2 3

, Lb =

1 2 3 42 1 4 33 4 1 24 3 2 1

,

Lc =

1 2 3 42 1 4 33 4 2 14 3 1 2

, Ld =

1 2 3 42 4 1 33 1 4 24 3 2 1

.

Permuting the last three rows of each of the four Latin square matrices

La,Lb,Lc,Ld generates 6 Latin squares, and so there are 24 so-called standard-

form 4 × 4 Latin squares in all. By standard-form we require that the top

row be 1, 2, 3, 4 but not necessarily the first column.


We identify these 24 standard-form Latin squares with a subscript letter

a, b, c, d and a superscript set of three numbers as they appear in column

1, rows 2,3,4. For example we represent the PLS (Figure 4.1, left panel) of

type a234 by

L(234)a =

1 2 3 42 3 4 13 4 1 24 1 2 3

.

10/05/2008 03:43 PMGoogle Image Result for http://store.coolstamps.com/imagesrv/532/926855148.jpg

Page 2 of 2http://images.google.com/imgres?imgurl=http://store.coolstamps.co…2BDisney%26um%3D1%26hl%3Den%26client%3Dsafari%26rls%3Den%26sa%3DN

CollectionsCoversDinosaursDisney

PocahontasAladdinAlice in WonderlandBambiCinderellaComic BooksDaisy DuckDisney:Donald DuckEarlyEasterFairy TalesFull Length MovieGoofyHerculesHuey Dewey LouieHunchbackLion KingLittle MermaidMickeyMinnie MouseNarnia Lion Witch

WardrobeOccupationsPoohProfessionsScroogeSleeping BeautySnow WhiteTiggerToy StoryToys

EventsLighthousesMilitaryMusicNaturePeopleSci-FiSpaceSportsSuppliesTransportationUnique Topics

Albania 1999: Mickey mouse Albania 2000: Donald duck

Figure 4.1: Albania 1999, Scott 2590 (Mickey Mouse) & 2007, Scott 2811 (Pink Panther).

We present examples of 10 different types of 4 × 4 PLS from 8 differ-

ent countries: Albania, Canada, Gambia, Guinea, Guinea-Bissau, Pitcairn

Islands, Portugal, and Tristan da Cunha issued from 1972–2009.

As of October 31, 2010, we have identified 172 PLS with precisely 100

issued for the WWF 1988–2009 from 73 countries and 33 from Macau (31

of type b324) but only of these 10 types (out of 24): The most popular (by

far) are:


• a234 (58): backwards circulant, block-Latin, symmetric

1 2 3 42 3 4 1

3 4 1 24 1 2 3

,

• a432 (40): forwards circulant, block-Latin

1 2 3 44 1 2 3

3 4 1 2

2 3 4 1

,

• b324 (54): Sudoku form, magic square

1 2 3 43 4 1 2

2 1 4 3

4 3 2 1

.

The sheetlet in Figure 4.1 (left panel) forms a PLS of type a234, which is

a one-step backwards circulant, and is 2× 2 block-Latin in that the top left

2 × 2 corner of four stamps is the same as that in the bottom right corner,

and the top right corner of four stamps is the same as that in the bottom

left corner. This seems to be the most popular type of PLS—we have found

58 examples as of October 31, 2010. The stamps were issued by Albania in

1999 and feature Mickey Mouse.

The sheetlet in Figure 4.1 (right panel) forms a PLS of type a432, which

is a one-step forwards circulant, which is also popular—we have identified

40 examples as of October 31, 2010. The stamps were issued by Albania in

1999 and feature the Pink Panther.

The matrix L(324)a and the array of 16 stamps in the Canada-1972 PLS

(Figure 4.2, left panel) are in Sudoku form in that the four stamps each

appear once in the top left, top right, bottom left, and bottom right 2 × 2

corners, and so the full Latin square is a solution to a 4 × 4 mini-Sudoku

puzzle; Yates [39] says that such Latin squares have “balanced corners”. For

more about Sudoku see, e.g., [1, 3, 4, 34, 35, 38].

The PLS in Figure 4.2 (right panel) is essentially the only one3 we have

found of type a342, which is also in Sudoku form, like the PLS of type a324

from Canada (Figure 4.2, left panel). Depicted is the subantarctic fur seal

(Arctocephalus tropicalis). The stamps were issued in 2004 by Tristan da

Cunha, a group of islands in the south Atlantic.

The sheetlet shown in Figure 4.3 forms a PLS of type b324, which is

in Sudoku form and forms a magic square; this type is very popular. This

sheetlet was issued by Canada in 2005 and features 4 Canadian bridges as

described in detail on the sheetlet-verso (Figure 4.4): The Jacques Cartier

Bridge across the St. Lawrence River, the Angus L. Macdonald Bridge across

Halifax Harbour, the Canso Causeway connecting the Nova Scotia mainland

with Cape Breton Island and the Swinging Bridge in Souris, Manitoba.

3We have found 3 more PLS of type a342 embedded in larger sheets of stamps [16].


Figure 4.2: Canada 1972, Scott 582–585; (right) Tristan da Cunha 2004, Scott 747–750.

Figure 4.3: Canada 2005, Scott 2100–2103.

Figure 4.4: Canada 2005, Scott 2100–2103 (verso).


Figure 4.5: Guinea-Bissau 2002, Groth4 304; (right) Portugal 1997, Scott 2153–2156.

The PLS in Figure 4.5 (left panel), which is the only one of type b342 that

we have identified, is also in Sudoku form and also forms a magic square,

this time with both diagonals “Latin” in that each stamp appears once in

the main forwards and backwards diagonals. Depicted is the African buffalo

(Syncerus caffer), a large African cloven-hoofed mammal.

The sheetlet in Figure 4.5 (left panel) was issued by Guinea-Bissau (Guine-

Bissau), in western Africa; formerly Portuguese Guinea, upon independence,

it added the name of its capital, Bissau, to the country’s name in or-

der to avoid confusion between itself and the nearby Republic of Guinea

(Republique de Guinee), which was formerly French Guinea (and which is-

sued the PLS of type c423 shown in Figure 4.6).

This PLS of type b423 (Figure 4.5, right panel) also forms a magic square

with both diagonals “Latin” in that each stamp appears once in the main

forwards and backwards diagonals; the only other such type is b342 (Figure

4.5, left panel). Issued by Portugal in 1997, the stamps shown in Figure 4.5

(right panel) depict the Pyrenean desman (Galemys pyrenaicus), a small

semi-aquatic mammal that lives in the Iberian peninsula; its body is like

that of a muskrat, its nose like that of a hedgehog and its feet like that of

a duck-billed platypus. The desman uses its snout as a periscope to breathe

above the water’s surface.

These two PLS (Figure 4.5, the two panels), respectively of type b342

and b423, are orthogonal to each other in that the pair defines a philatelic

4Listed on the WWF Groth AG website [13] but apparently not listed by Scott [26].


“Graeco–Latin square” from which, using “Euler’s algorithm” Denes & Keed-

well [9, p. 212], we find1 2 3 43 4 1 24 3 2 12 1 4 3

+ 4

1 2 3 44 3 2 12 1 4 33 4 1 2

− 4

1 1 1 11 1 1 11 1 1 11 1 1 1

=

1 6 11 1615 12 5 28 3 14 910 13 4 7

which is a fully-magic matrix in “classic” form in that the numbers 1, 2, . . . , 16

each appear once.

Figure 4.6: Guinea 2000, Groth 275, PLS type c423.

The sheetlet in Figure 4.6 was issued by the Republic of Guinea (Republique

de Guinee). Depicted is the Guinea-baboon (Papio papio) in row 1, columns

1 and 2, and the collared mangabey (Cercocebus torquatus) in row 1, columns

3 and 4. The Guinea-baboon is a baboon from the Old World monkey fam-

ily with reddish brown hair and a hairless, dark-violet or black face. The

collared mangabey is a long-tailed and long-legged, relatively large mon-

key with conspicuous white eyelids and short, backwardly-directed hairs on

the crown. This PLS is of type c423 and the only such PLS that we have

identified.

The sheetlet shown in Figure 4.7 (left panel) was issued by the Repub-

lic of The Gambia, commonly known as Gambia, the smallest country on

the African continental mainland. It is bordered to the north, east, and

south by Senegal, and has a small coast on the Atlantic Ocean in the west.

Depicted is the Black Crowned Crane (Balearica pavonina), a bird in the

family Gruidae. This species and the closely related Grey Crowned Crane

(Balearica regulorum), which prefers wetter habitats for foraging, are the

only cranes that can nest in trees. This PLS is of type b432, which is in

Sudoku form and is block-Latin.


Figure 4.7: Gambia 2996, Scott 3014ad; (right) Pitcairn Islands 2007, Scott 617ad.

Our 10th and last example of a PLS (Figure 4.7, right panel) is of type

d324 and the only PLS of this type that we have found; we call this type

criss-cross since the same stamp appears in each cell in the main forwards

diagonal, and the same stamp appears in each cell in the main backwards

diagonal. The stamps were issued by the Pitcairn Islands, well known as the

home of the descendants of the Bounty mutineers and the accompanying

Tahitians. In the top row, left to right, are the black noddy (Anous minu-

tus), the brown noddy (Anous stolidus), the black-grey ternlet (Procelsterna

cerulea) and the sooty tern (Sterna fuscata).

4.1. 4 × 4 Rainbow Latin squares. The sheetlet of 16 stamps from Pak-

istan (Figure 4.8) is almost a PLS—even though 4 different water birds each

appear 4 times, the 4 copies of the stamp for each bird are not all identi-

cal. We may think of this as a one-step backwards circulant PLS with a

superimposed “rainbow-square” with each row defining (the 4 parts of) a

rainbow. We will call this a rainbow-PLS (R-PLS), more precisely a row-

rainbow-PLS (RR-PLS). Depicted are 4 kinds of water birds: (a) gadwall,

(b) common shelduck, (c) mallard, and (d) greylag goose. This is the only

RR-RLS that we have found.

We have, however, found a column-rainbow-PLS (CR-PLS). Issued by

Nepal in 2005, this PLS (Figure 4.9) is of the very rare fully-magic PLS-

type b342 with the 4 stamps in each column marked by a different colour.

Depicted are 4 kinds of mammals: (a) gangetic dolphin, (b) Indian pangolin,

(c) Asiatic white elephant, and (d) clouded leopard.

PHILATELIC LATIN SQUARES 173 PHILATELIC LATIN SQUARES OF SIZE 4×4

FIGURE 3.1: Pakistan 1992: Scott 790a–790d.

8

Figure 4.8: Pakistan 1992, Scott 790ad.

Figure 4.9: Nepal 2005, Scott 762ad.


5. PLS with n = 5

We now look at PLS with n = 5 for which we have identified just 17 examples

(including 8 embedded in two panes each with 100 stamps) with just 4

different patterns. We have identified 5× 5 PLS from 6 “countries” (stamp-

issuing authorities): Canada (8), Pakistan (1), Swaziland (1), Transkei (5),

Turkey (1), and the USA (1). These 17 PLS are all of the circulant type,

either one- or two-step backwards (C1B, C2B) or forwards (C1F, C2F).

A 5 × 5 Latin square is said to be in standard-form when the first row is

1, 2, 3, 4, 5 and in reduced-form when it is in standard form and in addition

the first column is 1, 2, 3, 4, 5. From the well-known Statistical Tables by

Fisher and Yates [12, Table XV, p. 86], we find that there are 56 reduced-

form 5 × 5 Latin squares (in contrast to just 4 reduced-form 4 × 4 Latin

squares and just 1 reduced-form 3× 3 Latin square) yielding 24× 56 = 1344

possible distinct types of 5 × 5 PLS in standard-form. Of these 1344 types

we have identified PLS of just these 4 types: C1B, C1F, C2B, C2F:

C1B =

1 2 3 4 52 3 4 5 13 4 5 1 24 5 1 2 35 1 2 3 4

, C1F =

1 2 3 4 55 1 2 3 44 5 1 2 33 4 5 1 22 3 4 5 1

,

C2B =

1 2 3 4 53 4 5 1 25 1 2 3 42 3 4 5 14 5 1 2 3

, C2F =

1 2 3 4 54 5 1 2 32 3 4 5 15 1 2 3 43 4 5 1 2

.

5.1. One-step circulants: C1B, C1F. We have identified 5 PLS of the

one-step backwards circulant type C1B: 1 from Swaziland and 4 from Canada

and 10 of the one-step forwards circulant type C1F: 5 from Transkei, 1 from

Turkey and 4 from Canada. The 8 PLS from Canada are embedded in two

panes each of size 10 × 10.

A 5 × 5 PLS of type C1B was issued by Swaziland in 1982 for “Wildlife

Conservation”, the first of a series of 4 sets of five stamps, see Figure 5.1.

This first set was one of the earliest sets of stamps issued for the WWF.

Depicted in Figure 5.1 is Pel’s fishing-owl (Scotopelia peli), a large species

of owl which feeds nocturnally on fish snatched from the surface of lakes

and rivers. Printed on the stamps are the words Sikhova setinhlanti, which

is fishing-owl in SiSwati, a Bantu language spoken in Swaziland and South

Africa. Depicted are (top row left to right) male, female, pair, nest and egg,


adult (with youngster). The Kingdom of Swaziland is a landlocked country

in southern Africa, bordered by South Africa on three sides except to the

east, where it borders Mozambique.

Figure 5.1. Swaziland 1982, Scott 405ae, PLS type C1B. (Image courtesy Marc Steurs [28].)

Pel’s fishing owl is named after Hendrik Severinus Pel (1818–1876), who

was a member of the technical staff of the Rijksmuseum van Natuurlijke His-

torie in Leiden from 1832–1840 and then worked for the Dutch government

of the Gold Coast with the understanding that in his spare time he would

collect natural history specimens for the Leiden Museum. In the mid-19th

century, The Netherlands occupied six trade posts in what is now Ghana.


These stamps from Swaziland (Figure 5.1) seem to be quite rare. The

2010 Scott Standard Postage Stamp Catalogue lists a single stamp, unused,

at US$16.00 and a strip of five stamps, unused, at US$125.00, while on the

Steurs WWF website the full PLS is for sale at US$1609 on October 31, 2010

(US$1655.20 on January 29, 2010, up from US$1483.10 on June 12, 2008).

Delcampe [8] has a strip of 5 stamps (row 2) for sale at 62 Euros (US$86.46)

on October 31, 2010, while the Groth WWF website lists these stamps as a

“forerunner” (Groth #F40) of those issued for the WWF and lists a strip

of 5 stamps for sale 150 Swiss francs (US$152.28 but not in stock).

Figure 5.2. Turkey 1960, Scott 1488–1492, PLS type C1F.

A 5 × 5 PLS of type CF1 was issued in 1960 by Turkey (shown in Figure

5.2), the earliest 5 × 5 PLS that we have identified (the earliest 2 × 2 PLS

that we have found was issued by Canada in 1960). This PLS was issued

for the 1960 Summer Olympics held in Rome, Italy, August 25–September

11, 1960. Rome had been awarded the organization of the 1908 Summer

Olympics, but after the 1906 eruption of Mount Vesuvius, was forced to

decline and pass the honors to London. At the 1960 Olympics, Turkey won

7 gold medals and 2 silver—all 9 for wrestling. Depicted on the stamps are

(top row, left to right): hurdling, soccer, steeplechase, basketball, wrestling.


5.2. 5×5-embedded PLS.

Figure 5.3. Canada 1970, Scott 519–523.

Four 5× 5 PLS are embedded in each of two panes of 100 stamps, one pane

with 5-cent stamps (Figure 5.3), the other with 6-cent stamps. These panes

of stamps were issued by Canada for Christmas 1970 featuring children’s

designs. The pattern in each of the two panes is the same:(A AF

FA FAF

)=

(I

F

)A(I F

), F =

0 0 0 0 10 0 0 1 0

0 0 1 0 00 1 0 0 01 0 0 0 0

.

Here the 5 × 5 matrix A denotes the one-step backwards circulant matrix

C1B, the matrix I the 5×5 identity matrix, and F the 5×5 flip matrix. The

PLS defined by AF (top right corner) and FA (bottom left corner) are one-

step forwards circulants of type C1F and FAF (bottom right corner) is a

one-step backwards circulant of type C1B like the PLS defined by A (top left

corner), but with the stamps arranged differently. There are, therefore, eight

distinct PLS in all, with two one-step forwards circulants and two one-step

backwards circulants in each of the two panes. An interesting feature of these

stamps is that just one of the five stamps appears four times contiguously

in a block of four and so only in the centre of the full pane (Figure 5.3).


Figure 5.4. Pakistan 2004, Scott 1045ae; PLS type C2B.

5.3. Two-step circulants: C2B, C2F. The only 5 × 5 PLS of the two-

step circulant types C2B and C2F that we have identified are from Pakistan

(Figure 5.4) and the USA (Figure 5.5). These PLS have several interesting

properties. In particular they are both magic-Latin in that each of the 5

stamps appears once in the two main diagonals. Moreover, [23]: “Of Latin

squares used for crop experiments with a single set of treatments, the earliest

examples (published in 1924 [36]) are 5 × 5 squares of the types [C2B and

C2F] known as Knut Vik or “knight’s move” designs; these are squares where

all cells containing any one of the treatments can be visited by a succession

of knight’s moves (as in chess) and where no two diagonally adjacent cells

have the same treatment.”

As noted by Fisher in The Design of Experiments [11, p. 78]: “In this

arrangement the areas bearing each treatment are nicely distributed over

the experimental area, so as to exclude all probability that the more impor-

tant components of heterogeneity should influence the comparison between

treatments.” In 1931 Tedin [33] and in 1951 Nissen [21] observed that there

are just two possible arrangements of the 5 × 5 Knut Vik design in standard-

form: C2B and C2F. These designs are pandiagonal, in that all diagonals

(the two main diagonals and all broken diagonals) contain each of the five

treatments precisely once.


Figure 5.5. USA 1995, Scott 3019v; PLS type C2F.

A PLS of type C2B was issued by Pakistan (Figure 5.4) for the Na-

tional Philatelic Exhibition in Lahore on Universal Postal Union Day (Oc-

tober 9, 2004). This PLS depicts five kinds of tropical fish (top row, left

to right): striped gourami (Colisa fasciata), black widow tetra (Gymnoco-

rymbus ternetzi), yellow dwarf cichlid (Apistogramma borellii), tiger barb

(Capoeta tetrazona), neon tetra (Paracheirodon innesi).

A PLS of type C2F was issued by the USA (Figure 5.5) depicting five

antique automobiles from the late 19th and (very) early 20th century. In

the top row (left to right): 1893 Duryea, 1898 Columbia, 1901 White, 1894

Haynes, 1899 Winton.

6. Acknowledgements

We are most grateful to Marc Steurs [28] for the image of the (very rare) PLS

in Figure 5.1; all other PLS images are from the personal collection [31] of the

second author. Information about philatelic Latin squares may be found in the

open-access websites [2, 8, 10, 13, 14, 19, 20, 28], as well as in the Scott stamp

catalogues [26]. Much of our biographical and geographical information is based on

that in Wikipedia [37]. Many thanks to S. Ejaz Ahmed, Nicolas Ammerlaan, Oskar

Maria Baksalary, Christian Boyer, Ka Lok Chu, Jeffrey J. Hunter, Ian D. Kim-

merly, Alexander Kovacec, Sid Kroker, Sunil Kukreja, Michael D. Larsen, Friedrich

Pukelsheim, Simo Puntanen, Daniel J. H. Rosenthal, Evelyn Matheson Styan, Gotz

Trenkler, and Kimmo Vehkalahti for their help. Research was supported, in part,

by the Natural Sciences and Engineering Research Council of Canada.


References

[1] R. A. Bailey, Peter J. Cameron & Robert Connelly (2008). Sudoku, gerechte de-signs, resolutions, affine space, spreads, reguli, and Hamming codes. The AmericanMathematical Monthly, 115 (5), 383–404. 12

[2] BiStamp.com. Stamp professionals specializing in postage stamps of Russia and otherex-USSR states and worldwide topicals: online open-access website at Kyiv, Ukraine.6, 23

[3] Christian Boyer (2007). Sudoku’s French ancestors. The Mathematical Intelligencer,29 (1), 37–44. [Revised and enhanced version (in English) of the original Frenchversion: “Les ancetres francais du sudoku” in Pour La Science, no. 344, pp. 8–11 &89, June 2006.] 12

[4] Christian Boyer (2007). Sudoku’s French ancestors: solutions to problems. The Math-ematical Intelligencer, 29 (2), 59–63. 12

[5] Ka Lok Chu, Simo Puntanen & George P. H. Styan (2009). Some comments onphilatelic Latin squares from Pakistan. Pakistan Journal of Statistics, 25 (4), 427–471: online open-access. 1

[6] Charles J. Colbourn, Jeffrey H. Dinitz & Ian M. Wanless (2007). Latin squares. InHandbook of Combinatorial Designs Charles J. Colbourn & Jeffrey H. Dinitz, eds.),Chapman & Hall/Taylor & Francis, Boca Raton, pp. 135–152 (§ III, ch. 1). 1

[7] [Francois] Crette de Palluel (1788). Sur les avantages & l’economie que procurentles racines employees a l’engrais des moutons a l’etable. Memoires d’agriculture,d’economie rurale et domestique, Trimestre d’ete 1788 (“Lu le 31 juillet 1788”),pp. 17–23 (& 2 foldout-pages of tables facing p. 23). 2

[8] Sebastien Delcampe: delcampe.com. The international marketplace for [stamp] col-lectors: online open-access website in Soignies, Belgium. 20, 23

[9] J. Denes & A. D. Keedwell (1974). Latin Squares and their Applications. AcademicPress, New York and Akademiai Kiado, Budapest. 15

[10] eBay.com. Open-access auction and shopping website: online for stamps. 23[11] R. A. Fisher (1990). Statistical Methods, Experimental Design, and Scientific Infer-

ence, a re-issue of three books in one of: (book 1) Statistical Methods for ResearchWorkers, (book 2) The Design of Experiments, and (book 3) Statistical Methods andScientific Inference, edited by J. H. Bennett. Oxford University Press. [Reprintedwith corrections 1995.] 22

[12] Sir Ronald A. Fisher & Frank Yates (1963). Statistical Tables for Biological, Agri-cultural and Medical Research, Sixth Edition, Revised and Enlarged. Oliver & Boyd,Edinburgh; online open-access: R. A. Fisher Digital Archive, Adelaide Research andScholarship. 18

[13] Groth AG. WWF Conservation Stamp Collection: online open-access website inUnterageri (canton Zug), Switzerland. 5, 14, 23

[14] Library and Archives Canada (2008). Canadian Postal Archives Database: onlineopen-access search by keyword or by year. 23

[15] [Ramon Llull (1232–1316)] (2007). Raimundi Lulli Opera Latina 27: Ars Demon-strativa, edited by (cura e studio) Josep Enric Rubio Albarracın. Text in Latin withintroductions in Spanish and foreword in German. Corpus Christianorum ContinuatioMediaevalis 213, Raimundi Lulli opera Latina 32. Brepols, Turnhout. 2

[16] Peter D. Loly & George P. H. Styan (2009). Philatelic Latin squares. Report 2009-01,Department of Mathematics and Statistics, McGill University, Montreal. 1, 12

[17] Peter D. Loly & George P. H. Styan (2010). Comments on 4 × 4 philatelic Latinsquares. Chance, 23 (1), 57–62 (Winter 2010). 1

[18] Peter D. Loly & George P. H. Styan (2010). Comments on 5 × 5 philatelic Latinsquares. Chance, 23 (2), 58–62 (Spring 2010). 1

[19] Marlen Stamp & Coins Ltd: online open-access website in Great Neck, New York. 23[20] B. McLean Stamps for Collectors: P.O. Box 323, Ellon, Aberdeenshire AB41 7YA,

Scotland UK. online open-access website. 3, 23

http://www.bistamp.com/

http://www.pakjs.com/journals//25(4)/25(4)6.pdf

http://stamps.delcampe.com/page/main/language,E.html

http://desc.shop.ebay.com/Stamps__W0QQ_sacatZ260?_npmv=3

http://digital.library.adelaide.edu.au/dspace/handle/2440/10701

http://www.wwfstamp.com/

http://www.collectionscanada.gc.ca/archivianet/02011702_e.html

http://www.collectionscanada.gc.ca/archivianet/02011703_e.html

http://www.marlen-stamps.com/

http://www.bmclean.co.uk/


[21] Øivind Nissen (1951). The use of systematic 5×5 squares. Biometrics, 7 (2), 167–170;online at JSTOR. 22

[22] Lewis P. Orans (2004). Reverend Professor Baden Powell, Savilian Professor of Ge-ometry at Oxford: online open-access. 2

[23] D. A. Preece (1990). R. A. Fisher and experimental design: a review. Biometrics, 46(4), 925–935: online at JSTOR. 2, 22

[24] Terry Ritter (2003). Latin squares: a literature survey. Research comments from Ci-phers By Ritter: online open-access, last updated: 2003-12-21 (from 1998-05-29). 1

[25] William L. Schaaf (1978). Mathematics and Science: An Adventure in PostageStamps. National Council of Teachers of Mathematics, Reston, Virginia. 1

[26] Scott Standard Postage Stamp Catalogue (James E. Kloetzel, ed.), currently publishedannually in several volumes by Scott Publishing, Sidney, Ohio. 2, 4, 14, 23

[27] David Curtis Skaggs (1978). Postage stamps as icons. In Icons of America (Ray B.Browne & Marshall Fishwick, eds.), Popular Press, Bowling Green, Ohio, pp. 198–208.1

[28] Marc Steurs. The most complete site for your WWF stamp collection: online open-access website at Rijmenam (near Antwerp), Belgium. 5, 19, 23

[29] George P. H. Styan (2007). A philatelic introduction to magic squares and Latinsquares for Euler’s 300th birthyear. In Proceedings of the Canadian Society for Historyand Philosophy of Mathematics, vol. 20, pp. 306–319. 1

[30] George P. H. Styan (2009). An illustrated philatelic introduction to 4×4 Latin squaresin Europe: 1283–1788 (with some comments about le fauteuil Dagobert and aboutphilatelic Latin squares in Europe: 1984–2007). Talk presented at the 18th Interna-tional Workshop on Matrices and Statistics: IWMS’09, Smolenice Castle, Smolenice,Slovakia, 26 June 2009, overheads pdf file, 28 pp. 1

[31] George P. H. Styan (2010). Personal collection of philatelic items. Verdun (Quebec),Canada, October 31, 2010. 23

[32] George P. H. Styan, Christian Boyer & Ka Lok Chu (2009). Some comments on Latinsquares and on Graeco-Latin squares, illustrated with postage stamps and old playingcards. Statistical Papers, 50 (4), 917–941: online at SpringerLink.1

[33] Olof Tedin (1931). The influence of systematic plot arrangement upon the estimateof error in field experiments. The Journal of Agricultural Science, 21, 191–208. 22

[34] Peter Ullrich (1999). An Eulerian square before Euler and an experimental designbefore R. A. Fisher: on the early history of Latin squares. Chance, 12 (1), 22–26. 12

[35] Peter Ullrich (2002). Officers, playing cards, and sheep: on the history of Euleriansquares and of the design of experiments. Metrika, 56 (3), 189–204: online at Springer-Link. 12

[36] Knut Vik (1924). Bedømmelse av feilene pa forsøksfelter med og uten malestokk.Meldinger fra Norges Landbrukshøgskole, 4, 129–181. 22

[37] Wikipedia, The Free Encyclopedia. Wikimedia Foundation: online open-access webarchive. 1, 2, 4, 23

[38] Robin Wilson (2006). How to Solve Sudoku: A Step-by-Step Guide. Sterling, NewYork. 12

[39] F. Yates (1939). The comparative advantages of systematic and randomized arrange-ments in the design of agricultural and biological experiments. Biometrika, 30 (3/4),440–466: online at JSTOR. 12

Peter D. Loly: Department of Physics and Astronomy, The University ofManitoba, Winnipeg, Manitoba, Canada R3T 2N2

E-mail address: [email protected]

George P. H. Styan: Department of Mathematics and Statistics, McGillUniversity, 805 ouest, rue Sherbrooke, Montreal (Quebec), Canada H3A 2K6

E-mail address: [email protected]

http://www.jstor.org/stable/pdfplus/3001744.pdf

http://www.pinetreeweb.com/bp-father.htm

http://www.jstor.org/stable/2532438

http://www.ciphersbyritter.com/RES/LATSQ.HTM

http://www.panda.to/

http://www.springerlink.com/content/d582p612738777lw/fulltext.pdf

http://www.springerlink.com/content/r9k0rye5jjanm2rq/fulltext.pdf

http://en.wikipedia.org

http://www.jstor.org/stable/2332658

philatelic latin squareshome.cc.umanitoba.ca/~loly/lolystyan3-cshpm.pdf · 1947) on 9 april 2005....

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