The Surprising Versatility of Edge-Matching Tiles

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Presentation transcript:

The Surprising Versatility of Edge-Matching Tiles

Grand Snowflake, Bridges 2015 art exhibit

MiniMatch-I 9 edge-colored squares Presentation by Kate Jones at Bridges 2017 Waterloo, Canada

Research Results: 1. Enclose any one color

2. Enclose two colors. Only purple/white and red/white are solvable.

3. Wrap-arounds. Cylinder. Torus. Moebius Strip. Klein Bottle 3. Wrap-arounds. Cylinder Torus Moebius Strip Klein Bottle Projective Plane

4. Wrap-around transformations 4. Wrap-around transformations. After shifting one row from top to bottom:

5. Unmatched row ends. Very difficult.

6. Interior color divisions 6. Interior color divisions. 12 interior color squares can contain 3 or 4 colors in all these different sums: 5 4 3 0 5 4 2 1 5 3 3 1 5 3 2 2 4 4 4 0 4 4 3 1 4 4 2 2 4 3 3 2 3 3 3 3 The only combinations with no solutions are 5 5 2 and 5 5 1 1

7. Other observations. Only one of the four colors can be entirely on the border. Can form 78 other color-matched symmetrical shapes. Here’s one. Can form perfect mirror symmetry of all colors, producing double wrap-arounds as well.

8. Non-matching arrangements. A liberating, artistic exercise…

9. Mega-constructions Any single 3x3 solution can tile infinitely—on floors, walls, as art.

There is no end! www.gamepuzzles.com