docIt Figures: Logic Puzzles Powered by Geometry
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It Figures: Logic Puzzles Powered by Geometry
It Figures: Logic Puzzles Powered by Geometry Jeffrey Wanko [email protected] Greg Hawk [email protected] Miami University Oxford, OH Presented at the NCTM Annual Meeting April 14, 2011 Indianapolis, Indiana 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? Shikaku J. Wanko & G. Hawk – Miami University It Figures - 1 NCTM 2011 Annual Meeting Shikaku Puzzle 1 (10x10) Shikaku Puzzle 2 (10x10) Shikaku Puzzle 3 (10x10) Shikaku Puzzle 4 (10x10) Shikaku Puzzle 5 (10x10) Shikaku Puzzle 6 (10x10) 2 - It Figures NCTM 2011 Annual Meeting Shikaku J. Wanko & G. Hawk – Miami University Shikaku Puzzle 7 (18x10) Shikaku Puzzle 8 (18x10) Shikaku Puzzle 9 (18x10) Shikaku J. Wanko & G. Hawk – Miami University It Figures - 3 NCTM 2011 Annual Meeting Shikaku Puzzle 10 (24x14) 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 4 - It Figures NCTM 2011 Annual Meeting Shikaku J. Wanko & G. Hawk – Miami University TENTAI SHOW Tentai Show puzzles were created by Nikoli. Other names that are used for these puzzles and their variations are Galaxies, Spiral Galaxies, and Sym-a-Pix. “Ten” is Japanese for dot while “tai-Show” means symmetry. “Tentai” is used to reference astronomical objects. Combining these elements produces the name “Tentai Show” which has the double meaning of rotational symmetry and an astronomical display. In a Tentai Show puzzle, a rectangular grid is shown with some circles placed on the grid. Some puzzles use only one color of circles (usually white) while others use black and white circles (the standard for Tantai Show). Some designers have created variations with more than two colors. The colors of the circles do not affect the basic goal of the puzzles—to subdivide the entire starting puzzle along grid lines so that each piece has 180˚ rotational symmetry. In addition, a circle must appear at the center of rotation for each piece (see examples and non-examples below). Tentai Show puzzles with two or more colors of circles have the added feature of producing a picture or design when each piece is filled with the color of the center circle. Example Pieces (180˚ rotational symmetry with circle at center) Non-Example Pieces All puzzles featured here come from Nikoli puzzle books Tentai Show 1 and Tentai Show 2. Tentai Show Example Tentai Show J. Wanko & G. Hawk – Miami University Tentai Show Example Solution It Figures - 5 NCTM 2011 Annual Meeting Tentai Show Puzzle 1 Tentai Show Puzzle 2 Tentai Show Puzzle 3 Tentai Show Puzzle 4 Tentai Show Puzzle 5 Tentai Show Puzzle 6 Tentai Show Puzzle 7 Tentai Show Puzzle 8 6 - It Figures NCTM 2011 Annual Meeting Tentai Show J. Wanko & G. Hawk – Miami University Tentai Show Puzzle 9 Tentai Show Puzzle 10 Tentai Show Puzzle 11 Tentai Show Puzzle 12 Tentai Show J. Wanko & G. Hawk – Miami University It Figures - 7 NCTM 2011 Annual Meeting Tentai Show Puzzle 13 Tentai Show Resources: • Nikoli invented Tentai Show puzzles. These can be found occasionally in their puzzle magazines or in Tentai Show books available at www.nikoli.co.jp/howtoget-e.htm • OnlineMathLearning.com (http://interactive.onlinemathlearning.com/fun_galaxies.php) features an interactive applet for solving puzzles online. Puzzles are randomly generated for a given size or one that you select. Puzzles do not create a picture. • Conceptis Puzzles (http://www.conceptispuzzles.com/index.aspx?uri=puzzle/sym-a-pix) features an interactive applet for solving puzzles online. New puzzles appear each week and create pictures with multiple colors. A few are free, the rest can be purchased. 8 - It Figures NCTM 2011 Annual Meeting Tentai Show J. Wanko & G. Hawk – Miami University 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 2 3 4 5 – – – – – Quadrilateral Parallelogram Rectangle Isosceles right triangle Rectangle Shapedoku Example 2 – Parallelogram 4 – Isosceles right triangle Shapedoku J. Wanko & G. Hawk – Miami University 3 – Rectangle 5 – Rectangle Shapedoku Example Solution 1 – Quadrilateral It Figures - 9 NCTM 2011 Annual Meeting 1 – Isosceles right triangle 2 – Isosceles right triangle 3 – Rhombus 4 – Isosceles right triangle 1 – Isosceles triangle 2 – Isosceles right triangle 3 – Rhombus 4 – Square Shapedoku Puzzle 1 Shapedoku Puzzle 2 1 2 3 4 5 – – – – – Quadrilateral Rectangle Isosceles trapezoid Parallelogram Isosceles triangle Shapedoku Puzzle 3 1 2 3 4 5 – – – – – Parallelogram Square Quadrilateral Parallelogram Isosceles trapezoid Shapedoku Puzzle 4 10 - It Figures NCTM 2011 Annual Meeting Shapedoku J. Wanko & G. Hawk – Miami University 1 – Square 2 – Rectangle 3 – Rhombus 4 – Parallelogram 5 – Right triangle Shapedoku Puzzle 5 1 2 3 4 5 – – – – – Square Isosceles trapezoid Parallelogram Rectangle Isosceles trapezoid Shapedoku Puzzle 6 1 2 3 4 5 – – – – – Right trapezoid Square Parallelogram Isosceles triangle Rectangle Shapedoku Puzzle 7 Shapedoku J. Wanko & G. Hawk – Miami University It Figures - 11 NCTM 2011 Annual Meeting 1 – Kite 2 – Right trapezoid 3 – Square 4 – Right triangle 5 – Isosceles trapezoid Shapedoku Puzzle 8 1 2 3 4 5 6 – – – – – – Isosceles trapezoid Parallelogram Parallelogram Parallelogram Square Square Shapedoku Puzzle 9 1 2 3 4 5 6 – – – – – – Square Kite Rectangle Parallelogram Parallelogram Isosceles trapezoid Shapedoku Puzzle 10 12 - It Figures NCTM 2011 Annual Meeting Shapedoku J. Wanko & G. Hawk – Miami University 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 Rep-tiles J. Wanko & G. Hawk – Miami University Sphinx rep-4 It Figures - 13 NCTM 2011 Annual Meeting 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 14 - It Figures NCTM 2011 Annual Meeting Rep-tiles J. Wanko & G. Hawk – Miami University 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. Rep-tiles J. Wanko & G. Hawk – Miami University It Figures - 15 NCTM 2011 Annual Meeting 16 - It Figures NCTM 2011 Annual Meeting Rep-tiles J. Wanko & G. Hawk – Miami University SOLUTIONS Shikaku Puzzle 1 Puzzle 2 Puzzle 5 Puzzle 6 Puzzle 3 Puzzle 4 Puzzle 7 Puzzle 8 Puzzle 9 Puzzle 10 Puzzle 11 Solutions J. Wanko & G. Hawk – Miami University It Figures - 17 NCTM 2011 Annual Meeting SOLUTIONS Tentai Show Puzzle 1 Puzzle 2 Puzzle 3 Puzzle 4 Puzzle 5 Puzzle 6 Puzzle 7 Puzzle 8 Puzzle 9 Puzzle 10 Puzzle 11 Puzzle 12 18 - It Figures NCTM 2011 Annual Meeting Puzzle 13 Solutions J. Wanko & G. Hawk – Miami University SOLUTIONS Shapedoku Puzzle 1 Puzzle 2 Puzzle 5 Puzzle 8 Solutions J. Wanko & G. Hawk – Miami University Puzzle 3 Puzzle 6 Puzzle 9 Puzzle 4 Puzzle 7 Puzzle 10 It Figures - 19 NCTM 2011 Annual Meeting 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 20 - It Figures NCTM 2011 Annual Meeting Solutions J. Wanko & G. Hawk – Miami University
