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## How My Students Proved Me Wrong!

Two men, X and Y, meet on the street. X says: "All three of my sons have their birthday this very day."

Y asks: "How old are they?"

X replies: "The product of the ages of my sons is 36."

X points at the house next to them: The sum of their ages is equal to the number of windows you see in this building."

"My oldest son has blue eyes" says X.

"This is sufficient" exclaims Y and gives the correct ages of the sons.

Can you?

Note: When given this problem, my students creatively proved that the "expected" answer is incorrect...and that more information is needed...do you see why?

Source: Z. Michalevicz & D. Fogel's How To Solve It: Modern Heuristics, 2000.

Hint: Use common sense...take each clue at a time, but pay attention to the story's sequence.

Solution Commentary: The "expected" solution: Since the product of the three "integer" ages is 36, the eight possibilities are {1, 1, 36}, {1, 2, 18}, {1,3,12}, {1,4,9}, {1,6,6}, {2,2,9}, {2,3,6}, and {3,3,4}. For these possibilities, the respective sums are 38, 21, 16, 14, 13, 13, 11, and 10. Y remains confused due to the two sums 1+6+6 = 2+2+9 = 13, which equals the number of windows. Because there is an older son, the solution must be 2, 2, and 9.

Now, what concern did my students creatively raise...to derail this "expected"? They said that whenever twins are born, one must be older than the other, and thus, the ages of 1, 6, 6 remain a possibility....Ah, the delight of working with creative students!