challenge your mind: puzzles for your...

Challenge Your Mind: Puzzles for Your Classroom

Jeffrey Wanko [email protected]

Miami University

Oxford, OH

Greg Hawk [email protected]

Piqua High School

Piqua, OH

Presented at Ohio Council of Teachers of Mathematics Annual Conference – Columbus, OH

October 18, 2012

Shikaku Challenging Your Mind – 1 J. Wanko & G. Hawk OCTM

SHIKAKU Shikaku puzzles were created by the Japanese puzzle magazine Nikoli. They have also been published in the United States as Partitions puzzles. “Shikaku ni kire” is Japanese for “divide by squares” or “divide by box”, indicative of the broad goal of these puzzles. In a Shikaku puzzle, a rectangular grid is shown with white numbers placed in black circles in various squares throughout the grid. The goal of the puzzle is to divide the grid into rectangles and squares – each containing exactly one circled number – such that the area of each rectangle is the circled number it contains. Each square of the grid must be included in exactly one rectangle/square; in other words, every grid square must be used but no rectangles may overlap. Each puzzle has exactly one correct solution. All puzzles featured here come from the Nikoli website (see the resources section).

Shikaku Example

Shikaku Example Solution What strategies might you use to begin solving? What would make one Shikaku puzzle challenging compared to another one? What ideas regarding mathematics and geometry does this puzzle develop? How might you use these puzzles in your own classroom?

2 – Challenging Your Mind Shikaku OCTM J. Wanko & G. Hawk

Shikaku Puzzle 1 (10x10)




Shikaku Puzzle 5 (10x10) Shikaku Puzzle 6 (10x10)

Shikaku Challenging Your Mind – 3 J. Wanko & G. Hawk OCTM




4 – Challenging Your Mind Shikaku OCTM J. Wanko & G. Hawk


Shikaku Puzzle 11 (24x14) Shikaku Resources:

• www.nikoli.com – website of Nikoli (the Japanese puzzle magazine that invented Shikaku) which includes ten sample hand-made Shikaku puzzles that can be solved online. Additional puzzles can be played with a paid membership.

• www.puzzle-shikaku.com – millions of computer-generated Shikaku puzzles that can also be printed out to solve on paper

• www.shikakuroom.com – a puzzle generator that will create Shikaku puzzles from 2x2 to 20x20

Rep-Tiles Challenging Your Mind – 5 J. Wanko & G. Hawk OCTM

REP-TILES In 1962, Solomon Golomb began exploring shapes that could be used to create larger and smaller copies of themselves. He named these shapes “replicating figures” or “rep-tiles” and thus began an interesting geometric study that builds on the idea of similarity. Rep-tiles are related to tessellations (tilings), but they are different in one important factor. Rep-tiles are a subset of tessellations—that is, all rep-tiles are tessellations but not all tessellations are rep-tiles. For example, every triangle both tessellates and is a rep-tile because copies of a triangle can be combined to make a larger, similar copy of the same triangle (See below). On the other hand, the regular hexagon tessellates but is not a rep-tile because no number of tessellating hexagons will ever create a larger hexagon (see below).

Example Rep-tile Not a Rep-tile Another example of a rep-tile is the pentagon (known as the Sphinx) that is shown below. Using four copies of the same size, another Sphinx can be created. Because four copies can be used to create this rep-tile, it is called a rep-4 polygon.

Sphinx Sphinx rep-4

6 – Challenging Your Mind Rep-Tiles OCTM J. Wanko & G. Hawk

Each of the shapes below is also a rep-4 polygon. Can you find a way to fit together four copies of a shape to make a larger shape that is mathematically similar to the original shape? Can you find a way to fit together some other number of copies (n) to make a larger similar shape (to show that a shape is also rep-n)?

Small L Large L

Right Trapezoid Isosceles Trapezoid

P Pentomino

Rep-Tiles Challenging Your Mind – 7 J. Wanko & G. Hawk OCTM

Students might find it helpful to have a frame in which they could place copies of a shape to explore rep-tiles. Here are frames for the rep-4 explorations of the shapes from the previous page.

Shapedoku Challenging Your Mind – 9 J. Wanko & G. Hawk OCTM

SHAPEDOKU As with a Sudoku puzzle, the numbers in the Shapedoku puzzle solutions appear once in each row and each column. However, there are no outlined regions. Instead, clues are given describing the shape that would be created if the non-circled numbers of that type were connected (consider connecting the centers of the squares in which these numbers are placed). The shapes that are given are the most specific shape name for those numbers. For example—a shape described as a parallelogram will not be a rectangle, square, or rhombus. If it were, then the more specific name would be used. For example, in the 5 x 5 puzzle below, six circled numbers have been placed in the starting grid at the left. The remaining numbers must be placed so that they form the vertices of the shapes indicated. In the solution at the right, the numbers have been placed so that each number appears in each row and column, and so that the non-circled numbers of each type form the shapes that are indicated (see the three shape grids below).

1 – Quadrilateral 2 – Parallelogram 3 – Rectangle 4 – Isosceles right triangle 5 – Rectangle

Shapedoku Example

Shapedoku Example Solution

2 – Parallelogram

4 – Isosceles right triangle 3 – Rectangle 5 – Rectangle

1 – Quadrilateral


1 – Isosceles right triangle


3 – Rhombus 4 – Isosceles right

triangle

1 – Isosceles triangle


3 – Rhombus 4 – Square

Shapedoku Puzzle 1

Shapedoku Puzzle 2

1 – Quadrilateral 2 – Rectangle 3 – Isosceles trapezoid 4 – Parallelogram 5 – Isosceles triangle

Shapedoku Puzzle 3

1 – Parallelogram 2 – Square 3 – Quadrilateral 4 – Parallelogram 5 – Isosceles trapezoid

Shapedoku Puzzle 4

Shapedoku Challenging Your Mind – 11 J. Wanko & G. Hawk OCTM

1 – Square 2 – Rectangle 3 – Rhombus 4 – Parallelogram 5 – Right triangle

Shapedoku Puzzle 5

1 – Square 2 – Isosceles trapezoid 3 – Parallelogram 4 – Rectangle 5 – Isosceles trapezoid

Shapedoku Puzzle 6

1 – Right trapezoid 2 – Square 3 – Parallelogram 4 – Isosceles triangle 5 – Rectangle

Shapedoku Puzzle 7


1 – Kite 2 – Right trapezoid 3 – Square 4 – Right triangle 5 – Isosceles trapezoid

Shapedoku Puzzle 8

1 – Isosceles trapezoid 2 – Parallelogram 3 – Parallelogram 4 – Parallelogram 5 – Square 6 – Square

Shapedoku Puzzle 9

1 – Square 2 – Kite 3 – Rectangle 4 – Parallelogram 5 – Parallelogram 6 – Isosceles trapezoid

Shapedoku Puzzle 10

Solutions Challenging Your Mind – 13 J. Wanko & G. Hawk OCTM

SOLUTIONS Shikaku

Puzzle 1 Puzzle 2 Puzzle 3 Puzzle 4

Puzzle 5 Puzzle 6 Puzzle 7

Puzzle 8 Puzzle 9

Puzzle 10 Puzzle 11

14 – Challenging Your Mind Solutions OCTM J. Wanko & G. Hawk

SOLUTIONS Rep-tiles

Small L rep-4 Large L rep-4 Right Trapezoid rep-4

Small L rep-9 Large L rep-9 Right Trapezoid rep-9

Isosceles Trapezoid rep-4 P Pentomino rep-4

Isosceles Trapezoid rep-9 P Pentomino rep-9

Solutions Challenging Your Mind – 15 J. Wanko & G. Hawk OCTM

SOLUTIONS Shapedoku

Puzzle 1 Puzzle 2 Puzzle 3 Puzzle 4



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