Problem Solving Approach: The Hardest SAT Question Ever



As a tangible example of how the SAT® math section is different than a normal math test, allow me to share with you a notorious SAT® problem from the 1980’s and reproduced in a Scientific American article[1]:


“Place two quarters flat on a table so that they are touching. Holding one coin stationary on the table, roll the other quarter around it, keeping edge contact between the two without slipping. When the moving quarter returns to its starting location, how many full rotations has it made? In other words, how many times has George Washington returned to his upright position in the graphic below?”

This is an example of what the College Board, the writers and administrators of the SAT® college entrance exam, consider the hardest type of problem in their math section. Interestingly, while this problem can be addressed from a purely mathematical approach, including devising and writing an equation that describes and solves the problem, the problem is intended to be solved non-mathematically by building and testing mental models of what’s going on. Nor is this SAT problem a one-off. The hardest SAT math problems to this day continue to be of this type: more puzzle than math.

Take a moment for yourself to try and answer the question. Notice that any difficulty you, or I, or a student taking the SAT has with this problem, has nothing to do with any definitional problem or knowledge gap: there is no mathematical operation that one could learn to specifically make this problem easier.   Also notice what questions you are asking yourself, how you’re thinking about the question, the steps and ideas you take and have to move forward with the problem. This is an example of the explicit process that my students and I work through to figure out how to address the SAT® math questions that are more puzzle, or riddle, than math problem.. This is also exactly the type of problem solving approach that is difficult to cover and explore in traditional education environments and impossible with even the most cutting edge AI tools.

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