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One Student's Insight Trumps Twenty Students' Blinders

This problem has an interesting history. It was first used as part of a psychological exam....

A board was sawed into two pieces. One piece was two-thirds as long as the whole board and was exceeded in length by the second piece by 4 feet. How long was the board before it was cut?
Can you solve the problem? Algebra seems like an appropriate approach.

The reason the problem was part of a psychological exam was that psychologists wanted to see what a mathematically-competent person does when they solve a real-world problem that produces an impossible answer. That is, you should have gotten an answer of -12 feet!

The psychologists then watched the problem solvers' actions... avoidance... reworking... trying some approach other than "infallible" algebra, such as guess and test... asking for help...etc.

So, I gave this same problem to a group of 21 high-achievers to also see how they would handle the impossible answer. After reading the problem closely, your intuition should be triggered that a board two-thirds of the length cannot be exceeded in length by the other piece!

Out of about 21 secondary students, all but one smiled at the "false" answer and then went on to other problems. No consternation of any sort. No questioning as to the possibility of my wording the problem incorrectly, etc.

Then, about 30 minutes later, one student, the quietest of the bunch, raised his hand and stated he had solve the problem. Of course, he immediately had every one's attention, and they immediately raced back to try to resolve the problem. But, the problem-solvers were again disappointed...their algebra again failed them.

But, the lone student was correct. The problem, can be solved, with an infinite number of answers. DO YOU SEE HOW?


Hint: Draw a picture and set up a representative equation to solve.


Solution Commentary: I am unwilling to give the clever student's creative solution, but do suggest that you focus on how you draw the diagram to represent the problem. Put down your blinders and try to redraw the diagram in a multitude of ways consistent with the problem's premise.