triplewhist tournaments that are also mendelsohn designs

Triplewhist Tournaments That AreAlso Mendelsohn Designs

Ian Anderson,1 Norman J. Finizio2

1Department of Mathematics, University of Glasgow, Glasgow, Scotland G128QW

2Department of Mathematics, University of Rhode Island, Kingston, RI 02881,USA

Received March 27, 1996; accepted February 4, 1997

Abstract: We construct new families of whist tournaments that are at the same time bothtriplewhist tournaments and directedwhist tournaments. In particular, we construct such adesign on v elements whenever v is a product of primes pi, pi ≥ 29, pi ≡ 5 (mod 8). Itfollows that, for such v, two SOLSSOMs exist sharing the same mate. c© 1997 John Wiley &Sons, Inc. J Combin Designs 5: 397–406, 1997

Keywords: BIBD; whist tournaments; triplewhist tournaments; directedwhist tournaments; resolv-able perfect Mendelsohn designs

1. INTRODUCTION

A whist tournament Wh(4m + 1) for 4m + 1 players is a schedule of games (or tables)(a, b, c, d) involving two players a, c opposing two other players b, d, such that

i. the games are arranged into 4m+ 1 rounds each of m games;ii. each player plays in exactly in one game in all but one round;

iii. each player partners every other player exactly once;iv. each player opposes every other player exactly twice.

It may be helpful to think of (a, b, c, d) as the cyclic order of the 4 players sitting round atable. See [1, 2, and 10 (IV.53)] for basic information about whist tournaments.

We shall be concerned with two refinements of this structure, called triplewhist tourna-ments and directedwhist tournaments. Call the pairs {a, b} and {c, d} pairs of opponentsof the f irst kind, and call the pairs {a, d} and {b, c} pairs of opponents of the second

c© 1997 John Wiley & Sons, Inc. CCC 1063 8539/97/060397-10

397

398 ANDERSON AND FINIZIO

kind. Also say that b is a's left-hand opponent and c's right-hand opponent, and makesimilar definitions for each of a, c, and d. Then a triplewhist tournament TWh(4m + 1)is a Wh(4m + 1) in which every player is an opponent of the first (resp., second) kindexactly once with every other player; and a directedwhist tournament DWh(4m + 1) is aWh(4m+1) in which each player is a left (resp., right) hand opponent of every other playerexactly once. If the players are elements of Z4m+1, and if the ith round is obtained fromthe initial (first) round by adding i − 1 to each element (mod 4m + 1), then we say thatthe tournament is Z-cyclic. By convention we always take the initial round to be the roundfrom which 0 is absent. The games (tables)

(a1, b1, c1, d1), . . . , (am, bm, cm, dm)

form the initial round of a triplewhist tournament ifm⋃

i=1

{ai, bi, ci, di} = Z4m+1 − {0}, (α)

m⋃

i=1

{±(ai − ci),±(bi − di)} = Z4m+1 − {0}, (β)

m⋃

i=1

{±(ai − bi),±(ci − di)} = Z4m+1 − {0}, (γ)

m⋃

i=1

{±(ai − di),±(bi − ci)} = Z4m+1 − {0}, (δ)

whereas they form the initial round of a directedwhist tournament if (α), (β) hold alongwith

m⋃

i=1

{bi − ai, ci − bi, di − ci, ai − di} = Z4m+1 − {0}. (ε)

Example 1.1 A Z-cyclic DWh(13).

Round 1 (1, 8, 12, 5) (2, 3, 11, 10) (4, 6, 9, 7)Round 2 (2, 9, 0, 6) (3, 4, 12, 11) (5, 7, 10, 8)

......

......

Round 13 (0, 7, 11, 4) (1, 2, 10, 9) (3, 5, 8, 6).

The study of triplewhist tournaments has largely been separate from the study of directed-whist tournaments. Directedwhist tournaments are resolvable perfect Mendelsohn designsof block size 4; they appear to have been first studied by Baker and Wilson (see [6]).Triplewhist tournaments were first introduced by Moore [16], and recently there have beenmany articles on the construction ofZ-cyclic triplewhist tournaments [3, 4, 5, 12, 14]. Nowfor the first time we show that there are whist tournaments which are simultaneously bothtriplewhist and directedwhist tournaments. Such designs will be called directed triplewhisttournaments and will be denoted by DTWh(v). We shall show that a DTWh(v) exists forall sufficiently large v ≡ 1 (mod 4), and Z-cyclic ones exist whenever v is a product ofprimes pi ≥ 29, pi ≡ 5 (mod 8). Our methods will be similar to those used in [3], and willdepend on results on character sums which can be found in [15].

TRIPLEWHIST TOURNAMENTS 399

Finally, in Section 5, we shall show that the existence of a DTWh(v) implies the existenceof two different self-orthogonal Latin squares sharing the same symmetric orthogonal mate(SOLSSOMs).

2. A BASIC CONSTRUCTION

We are going to see how to construct the initial round of a Z-cyclic DTWh(p) where p isa prime ≡ 5 (mod 8). Let p = 8t+ 5, and let θ be a primitive root of p, so that θ8t+4 = 1and θ4t+2 = −1 in Zp. Consider the tables

(1, θ,−θ, θ−2) times 1, θ4, θ8, . . . , θ8t.

Because −θ = θ4t+3 and θ−2 = θ8t+2, we certainly have all the nonzero elements of Zp

here. The partner differences are

±(θ + 1),±θ−2(θ3 − 1) times 1, θ4, θ8, . . . , θ8t,

and these are all distinct provided exactly one of θ3−1, θ+1 is a square. The first opponentdifferences are

±(θ − 1),±θ−2(θ3 + 1) times 1, θ4, θ8, . . . , θ8t,

and these are all distinct provided exactly one of θ − 1, θ3 + 1 is a square. The secondopponent differences are

±2θ,±θ−2(θ2 − 1) times 1, θ4, θ8, . . . , θ8t,

and, since 2 is a nonsquare, these are all distinct provided θ2 − 1 is not a square.So the triplewhist conditions (α)–(δ) are satisfied provided

θ2 − 1 6= 2, θ2 + θ + 1 = 2, θ2 − θ + 1 = 2.

Now consider the requirement (ε) for a directedwhist tournament. The differences are

θ − 1,−2θ, θ−2(θ3 + 1), θ−2(θ2 − 1) times 1, θ4, θ8, . . . , θ8t.

If we set θ−1 = θa,−2θ = θb, θ+1 = θc and θ3 +1 = θd, then we obtain a directedwhisttournament provided a, b, d− 2 and a+ c− 2 are all distinct (mod 4).

Example 2.1 p = 29. Take θ = 19. Then a = 23, b = 12, c = 12, d = 16, θ2 − 1 =θ7 6= 2, θ2 + θ + 1 = 4 = 2, θ2 − θ + 1 = 24 = θ4 = 2. So we obtain

(1, 19, 10, 9) times 1, 194, . . . , 1924,

i.e.,

(1, 19, 10, 9), (24, 21, 8, 13), (25, 11, 18, 22), (20, 3, 26, 6),

(16, 14, 15, 28), (7, 17, 12, 5), (23, 2, 27, 4)

as the initial round of a Z-cyclic DTWh(29).This construction proves successful for all but a small number of cases. If we try it

for primes p ≡ 5 (mod 8), 29 ≤ p < 2000, we find that an appropriate primitive rootθp exists for all such primes except 37, 53, 61, 109, 149, 157, 229. We tabulate pairs(p, θp), p < 2000 in Table I.


TABLE I. (p, θp)

(29, 19), (37, –), (53, –), (61, –), (101, 26), (109, –), (149, –),(157, –), (173, 101), (181, 41), (197, 30), (229, –), (269, 39), (277, 18),(293, 116), (317, 45), (349, 71), (373, 5), (389, 34), (397, 217), (421, 18),(461, 46), (509, 42), (541, 67), (557, 47), (613, 60), (653, 18), (661, 88),(677, 17), (701, 66), (709, 22), (733, 133), (757, 158), (773, 34), (797, 285),(821, 53), (829, 390), (853, 208), (877, 105), (941, 95), (997, 387), (1013, 33),(1021, 287), (1061, 15), (1069, 43), (1093, 122), (1109, 43), (1117, 17), (1181, 95),(1213, 5), (1229, 19), (1237, 128), (1277, 28), (1301, 197), (1373, 35),(1381, 138), (1429, 79), (1453, 45), (1493, 20), (1549, 141), (1597, 85),(1613, 73), (1621, 17), (1637, 17), (1669, 17), (1693, 304), (1709, 19),(1733, 192), (1741, 98), (1789, 224), (1861, 11), (1877, 126), (1901, 232),(1933, 618), (1949, 28), (1973, 26), (1997, 18).

For the seven missing values of p = 8t+ 5, we can use the approach of Finizio [12] andconsider whist tournaments with initial round tables

(θ, 1, θ4k+2, θ4h+3) times 1, θ4, . . . , θ8t.

We find that the following choices of (p, θp, k, h) yield Z-cyclic DTWh(p):

(37, 2, 2, 5), (53, 2, 1, 9), (61, 2, 1, 6), (109, 6, 1, 17), (149, 2, 0, 24),

(157, 5, 4, 7), (229, 6, 0, 25).

Example 2.2 p = 37. Take θ = 2, k = 2, h = 5 to get initial round tables (2, 1, 210, 223)times 1, 24, . . . , 232, that is,

(2, 1, 25, 5), (32, 16, 30, 6), (31, 34, 36, 22), (15, 26, 21, 19),

(18, 9, 3, 8), (29, 33, 11, 17), (20, 10, 28, 13), (24, 12, 4, 23),

(14, 7, 27, 35).

Thus we have

Theorem 2.1. AZ-cyclic DTWh(p) exists for all primes p ≡ 5 (mod 8), 29 ≤ p < 2000.

The existence of a DTWh(29) and a DTWh(37) is enough to guarantee the existence ofa DTWh(v) for all sufficiently large v ≡ 1 (mod 4).

Theorem 2.2. A DTWh(v) exists for all sufficiently large v ≡ 1 (mod 4).

Proof. It follows from the results of Wilson [17] that a PBD({29, 37}, v) exists for allsufficiently large v ≡ 1 (mod 4). On each block B of this PBD, form a DTWh(|B|). Thenfor each x, take the blocks B containing x and, for each such B, take all the tables of theround of the DTWh on B in which x sits out. These games will form a round, which welabel as round x, of the required DTWh(v). It is clear that the triplewhist and directedwhistproperties are preserved by this construction.


3. THE EXISTENCE THEOREM

The construction given in Section 2 appears to produce a Z-cyclic DTWh(p) for primesp ≡ 5 (mod 8), p > 229, but proving that it will always work appears difficult. We thereforeturn to some other conditions which will guarantee the existence of a DTWh(p), and usecharacter sum arguments to show that the conditions can always be satisfied.

So let p = 8t+5 be prime, and let θ be a primitive root of p. We present six constructions.

Construction 1. (1, θ,−1, θ3) times 1, θ4, . . . , θ8t.

These are the initial round tables of a TWh(p) provided θ2 − 1 6= 2, θ2 ± θ + 1 = 2.They also yield a DWh(p) provided θ2 ± θ + 1 are both fourth powers.

Construction 2. (1, θ3, θ2,−θ3) times 1, θ4, . . . , θ8t.

These tables yield a TWh(p) provided θ2 − 1 6= 2, θ2 ± θ + 1 = 2 and they yield aDWh(p) provided θ2 ± θ + 1 are both squares but not fourth powers.

Construction 3. (1, θ3 − θ4,−θ3) times 1, θ4, . . . , θ8t.

For a TWh(p) we require θ4 +1 6= 2, θ2 ± θ+1 = 2. We also get a DWh(p) provided(θ − 1)/(θ + 1) is a fourth power and exactly one of θ2 ± θ + 1 is a fourth power.

Construction 4. (1, θ,−θ4,−θ) times 1, θ4, . . . , θ8t.

Here we need θ4 + 1 6= 2, θ2 ± θ + 1 = 2 for a TWh(p) and we also get a DWh(p)provided (θ − 1)/(θ + 1) is a square but not a fourth power and exactly one of θ2 ± θ + 1is a fourth power.

Construction 5. (1, θ,−θ4, θ3) times 1, θ4, . . . , θ8t.

For a TWh(p) we require θ2 − 1 = 2, θ4 + 1 = 2, (θ2 + θ+ 1)(θ2 − θ+ 1) = 2. Wealso get a DWh(p) provided θ2 + θ + 1 is a fourth power but θ2 − θ + 1 is not.

Construction 6. (1,−θ,−θ4,−θ3) times 1, θ4, . . . , θ8t.

As for Construction 5 except here we need θ2 − θ + 1 is a fourth power but θ2 + θ + 1is not.

We now state

Theorem 3.1. Let p = 8t+ 5 be prime. If there exists a primitive root θ of p such thatθ2 ± θ + 1 are both squares and either

θ2 − 1 6= 2 and (θ2 + θ + 1)(θ2 − θ + 1)

is a fourth power, or

θ2 − 1 = 2

and

(θ2 + θ + 1)(θ2 − θ + 1)

is not a fourth power, then a Z-cyclic DTWh(p) exists.


Proof. Suppose there exists such a primitive root θ. If it happens that θ2−1 is not a square,use Construction 1 if both θ2 ± θ+ 1 are fourth powers, and use Construction 2 otherwise.So suppose now that θ2 − 1 is a square. Next suppose θ4 + 1 6= 2. Since exactly one ofθ2 ± θ+1 is a fourth power, we can use Construction 3 if (θ− 1)/(θ+1) is a fourth powerand Construction 4 otherwise. Finally if θ2 − 1 = 2 and θ4 + 1 = 2, use Construction 5if θ2 + θ + 1 is a fourth power and use Construction 6 otherwise.

It therefore remains to show that a primitive root θ satisfying the conditions of Theorem3.1 can be obtained. We use the methods developed by Cohen [8, 9] and presented in anappropriate form by McNay [15] to obtain bounds on p beyond which the existence of sucha θ can be guaranteed.

By a character, we mean a homomorphism χ:Z∗p → C. The trivial character χ0 is

defined by χ0(x) = 1 for all x. The order of χ is the least positive integer d such thatχd = χ0, clearly d|(p− 1), and there are ϕ(d) characters of order d, where ϕ is the Eulerfunction.

Let λ denote the quadratic character mod p, so that λ(x) = 1 if x is a square (mod p),λ(x) = −1 if x is not a square. Let ψ be any fixed character of order exactly 4; thenψ(x) = 1 if x is a fourth power, and ψ(x) = −1 if x is a square but not a fourth power.Then

f(θ) = (λ(θ2 − θ + 1) + 1)(λ(θ2 + θ + 1) + 1)

× (1 − ψ((θ2 + θ + 1)(θ2 − θ + 1)(θ2 − 1)2)) (1)

takes the value 8 if θ satisfies the conditions of Theorem 3.1 and takes the value 0 otherwise.Now if µ denotes the Mobius function and, for d|(p− 1),

∑χd

denotes the sum over allϕ(d) characters of order exactly d, then [13] the expression

ϕ(p− 1)

p− 1

∑

d|(p−1)

µ(d)

ϕ(d)

∑

χd

χd(x)

takes the value 1 if x is a primitive root of p, and takes the value 0 otherwise. Thus thenumber of primitive roots of p which satisfy the conditions of Theorem 3.1 is

N =ϕ(p− 1)

8(p− 1)

∑

d|(p−1)

µ(d)

ϕ(d)

∑

x∈Zp

∑

χd

χd(x)f(x)

where f is given by (1). We want N > 0. Since λ(x) = ψ(x2) for each x, we can expand(1) to obtain

f(θ) = 1 +

7∑

j=1

ψ(Fj(θ))

for certain polynomials Fj , namely (θ2 + θ + 1)2, (θ2 − θ + 1)2, (θ2 − θ + 1)2(θ2 + θ +1)2,−(θ2 − θ + 1)(θ2 + θ + 1)(θ2 − 1)2,−(θ2 − θ + 1)3(θ2 + θ + 1)(θ2 − 1)2,−(θ2 −θ + 1)(θ2 + θ + 1)3(θ2 − 1)2,−(θ2 − θ + 1)3(θ2 + θ + 1)3(θ2 − 1)2.

Now, as shown in McNay's article [15], the bound on p, beyond which we can sayN > 0, depends on K, the sum of the degrees of the square free parts of the Fj . HereK = 2 + 2 + 4 + 6 + 6 + 6 + 6 = 32, so we read off from Table 2 of [15] that N > 0provided ω(p−1) > 7 (where ω(n) denotes the number of distinct prime factors of n). Wethen use the sieve argument of [15, Theorem 3] to deduce that N > 0 except possibly for


those p in the following sets:

S2 = {p ≤ 16, 384, ω(p− 1) = 2},S3 = {p ≤ 65, 536, ω(p− 1) = 3},S4 = {p ≤ 262, 144, ω(p− 1) = 4},S5 = {p ≤ 1, 035, 749, ω(p− 1) = 5},S6 = {p ≤ 2, 457, 009, ω(p− 1) = 6},S7 = {p ≤ 5, 542, 921, ω(p− 1) = 7}.

For the primes of interest here we find that |S2| = 94, |S3| = 749, |S4| = 2077, |S5| =2426, |S6| = 656, |S7| = 34.

We have checked by computer that appropriate primitive roots exist for all primes p ∈∪iSi except for 29, 61, 101, 157, 229. But DTWh(p) have already been constructed forthese p in Section 2. A printout of these results can be obtained from the authors. Sincethere are 23 pages of data, we limit ourselves here to presenting the results for S2 and S7. Ineach case we list (p, θp,m) where p is the prime, θp is the primitive root, andm, 1 ≤ m ≤ 6is the number of the construction used.

Primes in S2

(29, –, –), (37, 2, 4), (53, 14, 3), (101, –, –), (109, 14, 2), (149, 34, 4),

(173, 7, 1), (197, 12, 3), (269, 29, 2), (293, 8, 1), (317, 8, 6), (389, 3, 1), (509, 7, 1),

(557, 11, 2), (653, 12, 3), (677, 17, 2), (773, 12, 1), (797, 7, 1), (1109, 42, 1),

(1229, 17, 1), (1373, 12, 2), (1493, 11, 1), (1637, 41, 2), (1733, 32, 5), (1949, 27, 5),

(1997, 20, 3), (2309, 8, 2), (2477, 5, 3), (2693, 27, 2), (2837, 3, 1), (2909, 10, 3),

(2917, 52, 3), (2957, 61, 6), (3413, 8, 3), (3533, 11, 2), (3677, 8, 6), (3989, 44, 1),

(4133, 12, 1), (4157, 12, 2), (4253, 11, 2), (4349, 13, 5), (4373, 20, 2), (4493, 20, 6),

(4517, 17, 2), (5189, 15, 2), (5309, 11, 6), (5477, 32, 1), (5693, 3, 1), (5717, 17, 1),

(5813, 8, 1), (6173, 5, 4), (6197, 8, 4), (6269, 15, 1), (6317, 30, 2), (6389, 38, 4),

(6653, 5, 6), (7013, 5, 5), (7109, 27, 4), (7517, 8, 1), (7949, 12, 6), (8069, 17, 1),

(8117, 11, 2), (8573, 27, 1), (8837, 8, 1), (9173, 5, 3), (9533, 12, 2), (9749, 7, 2),

(10589, 46, 4), (10709, 39, 2), (10733, 12, 5), (10853, 30, 1), (11069, 18, 5),

(11213, 28, 3), (11549, 10, 1), (11813, 14, 3), (12149, 14, 4), (12197, 5, 6),

(12269, 48, 6), (12437, 31, 4), (12653, 14, 3), (12917, 31, 3), (13037, 42, 2),

(13229, 52, 2), (13829, 3, 2), (13877, 5, 4), (13997, 3, 1), (14549, 7, 1),

(14957, 27, 1), (15077, 11, 1), (15173, 27, 3), (15413, 29, 4), (15629, 35, 4),

(15773, 21, 2), (16229, 10, 6).

Primes in S7

(1381381, 130, 3), (1492261, 47, 4), (1741741, 2, 3), (1806421, 29, 2),

(1861861, 2, 6), (2134861, 2, 6), (2277661, 18, 5), (2582581, 2, 5), (2691781, 2, 5),

(2771341, 26, 6), (3354781, 28, 1), (3377221, 177, 5),

(3598981, 2, 5), (3663661, 2, 6), (3749461, 94, 1), (3805621, 72, 6),

(3838381, 2, 5), (3991261, 2, 6), (4024021, 2, 6), (4229941, 61, 5),


(4253341, 47, 6), (4264261, 10, 3), (4356661, 18, 4), (4476781, 10, 3),

(4744741, 2, 3), (4957261, 90, 4), (4994221, 2, 4), (5105101, 10, 4),

(5179021, 30, 4), (5262181, 2, 6), (5354581, 66, 3), (5399941, 17, 2),

(5476381, 2, 6), (5493181, 19, 1).

Thus the following theorem is established.

Theorem 3.2. A Z-cyclic DTWh(p) exists for all primes p ≡ 5 (mod 8), p ≥ 29.

If we now follow precisely the arguments of Section 3 of [3] we obtain the followingcorollary.

Theorem 3.3. If p1, . . . , pn are primes, pi ≥ 29, pi ≡ 5 (mod 8), then there exists aZ-cyclic DTWh(pα1

1 · · · pαnn ) for all n ≥ 1, αi ≥ 1.

Example 3.1 A DTWh(841). Here 841 = 292 and a DTWh(29) was obtained in Example2.1 with initial round games (1, 19, 1915, 1926) times 1, 194, . . . , 1924 in Z29. For aDTWh(841), we take as initial round tables

(a) 29(1, 19, 1915, 1926) times 1, 194, . . . , 1924, and(b) (1, 19, 1915, 1926) times 1, 194, . . . , 19836.

All arithmetic is carried out (mod 841).

4. THE THREE PERSON PROPERTY

A whist tournament is said to have the three person property if there is no set of 3 playerswho play together at more than one table, i.e., if the intersection of any two tables is alwaysat most 2. Whist tournaments with this property have been studied by Finizio [11]. Wenow show that there are infinitely many three person directed triplewhist tournaments.

Example 4.1 A Z-cyclic 3-person DTWh(29).

(1, 13, 15, 14), (24, 22, 12, 17), (25, 6, 27, 2), (20, 28, 10, 19),

(16, 5, 8, 21), (7, 4, 18, 11), (23, 9, 26, 3).

Example 4.2 A Z-cyclic 3-person DTWh(37).

(1, 2, 21, 35), (16, 32, 3, 5), (34, 31, 11, 6), (26, 15, 28, 22),

(9, 18, 4, 19), (33, 29, 27, 8), (10, 20, 25, 17), (12, 24, 30, 13),

(7, 14, 36, 23).

Theorem 4.1. There exists a 3-person DTWh(v) for all suf®ciently large v ≡ 1(mod 4).

Proof. As for Theorem 2.2.


5. QUASIGROUPS AND SOLSSOMS

A quasigroup is an ordered pair (Q, ?) where Q is a set and ? is a binary operation on Qsuch that the equations a?x = b and y?a = b are uniquely solvable for every pair a, b ∈ Q.The composition table of a quasigroup takes the form of a Latin square. A quasigroup isidempotent if x2 (=x ? x) = x for all x ∈ Q; it is a Schroder quasigroup if xy ? yx = xfor all x, y ∈ Q, and it satisfies Stein's Third Law if xy ? yx = y for all x, y ∈ Q. Thefollowing is well known [7].

Theorem 5.1. (i) If a TWh(v) exists, then an idempotent Schroder quasigroup of orderv exists.

(ii) If a DWh(v) exists, then a quasigroup satisfying Stein's Third Law exists.

The relevant constructions are (i) define a ? a = a and, if a 6= b, a ? b = c where aand c are opponents of the first kind when a partners b and (ii) define a ? a = a and, ifa 6= b, a ? b = c where (a, c, b, d) is a table.

The multiplication tables both give self-orthogonal Latin squares (SOLS). In both casesthe SOLS has an orthogonal mateM defined bymii = i,mij = number of round in whichi, j are partners (i 6= j).

Now when we have a whist tournament which is both a triplewhist and a directedwhisttournament, we can therefore obtain from it two different quasi-groups, one Schroder andthe other satisfying Stein's Third Law, and also two SOLS which share the same symmetricorthogonal mate, i.e., two different SOLSSOMs with the same mate. See [10 (IV.41)] formore information on SOLSSOMs.

6. CONCLUSIONS

We have shown that, for any prime p ≡ 5 (mod 8), p ≥ 29, there exists aZ-cyclic DTWh(p).We also remark that we have studied primes p = 2kt + 1, t odd, k ≥ 3. In particular,we have DTWh(41), DTWh(73), and DTWh(89). This study, as yet incomplete, will bereported on at a later date.

ACKNOWLEDGMENTS

We would like to thank Gavin McNay for his help, particularly with the material ofSection 3.

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