Elizabeth Ahlers
Grade 8, Georgetown Day School, Washington, DC

I figured this out mostly by trial and error but I had a plan so that I wouldn't make the same pattern. When ever I figured out a way to color in exactly half, I would flip it and rotate it and put those in a list of ways I couldn't do it. That kept me from making the same coloration twice. I thought about it for a long time but I was only able to come up with thirteen. To write out each one I have labeled the triangles and written for each way the number of the triangle colored in. The triangle directly above the horizontal dividing line is one, the next triangle over is two and continue moving counter clockwise. Here are the thirteen different ways that I got:

1. 1,2,3,4
2. 1,4,5,7
3. 1,3,5,7
4. 1,2,5,6
5. 2,3,6,7
6. 1,2,3,5
7. 2,5,7,8
8. 1,2,3,8
9. 1,2,4,5
10. 1,2,3,7
11. 1,2,3,6
12. 2,4,7,8
13. 1,4,6,7

Daniel Zainulbhai
Grade 8, Georgetown Day School, Washington, DC

I colored in half of the square 13 different ways. I just kept coloring the square different ways until I could find no more. I don't know exactly how much I had, but I had around 20. Then I drew each square on a separate piece of paper, and on both sides of the paper too. Then I compared each square to each other, and crossed the squares that were the same when flipped of rotated. I ended up with 13. I checked those thirteen again, to see if any were alike, but none were. Then I tried to see if I could find any more, but I couldn't. If you number the squares 1-8, I'll tell you the combinations I got. Label the square like this: 1,2,3,4
8,7,6,5

1,2,5,6
2,3,6,7
1,2,4,5
1,2,3,4
1,4,5,6
2,4,5,6
8,4,5,6
1,3,5,7
2,5,6,8
3,5,6,8
4,5,6,7
2,4,5,8
2,4,5,7

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