This was a fun problem, and I had to do it a couple of times before I came up with a good system. (I have been told that this is really a discrete math problem, but I figure since it uses a picture we can call it geometry!) The solutions chosen as winners and honorable mention all explain their system for figuring it out, which I think is important. An explanation should always give others some hints as to how they might go about solving a similar problem.

Winner: Elizabeth Ahlers, Grade 8, Georgetown Day School, Washington, DC
Winner: Daniel Zainulbhai, Grade 8, Georgetown Day School, Washington, DC
Elizabeth and Daniel both did a nice job of explaining how they narrowed it down to make sure they had all the solutions. Both strategies are based on trial and error, but as long as you're thorough, that works just fine! I especially like Daniel's idea of making squares of each design, and drawing on both sides of the paper so that you're sure to get more of the flips and rotations.

They also each came up with a good idea of communicating their solutions, by explaining how to number the triangles and then giving a list of the solutions. That can be tough with a problem like this when you're not sure how to submit pictures, but their system works really well.

Honorable Mention: Erin Kemp, Grade 10, Germantown Academy, Fort Washington, Pennsylvania
Honorable Mention: Jen Peters, Grade 10, Germantown Academy, Fort Washington, Pennsylvania
Erin and Jen both did a super job of explaining how they figured it out by breaking it down into different "groups" of colored triangles. They both looked at the different ways you could make "four" - as four, as three and one, and so on. They would have been the winners, since their explanations are a little more systematic than just guessing and checking, but they didn't figure out a way to communicate the answer without pictures.

We received more correct answers that didn't include any explanation at all - remember, the right answer is important, but so is the explanation. Can you really explain how you figured it out so that someone else could learn from reading your solution? That's always harder than finding the answer. Though we also received a number of wrong submissions to this problem! Checking for duplicates, or even knowing when you have all the answers, isn't easy unless you have a good system.

-Annie

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