Mind Boggling, Seemingly Impossible Age Problem
One Mathematician presents the following problem to another mathematician: "I have three daughters. The product of their ages is 72, and the sum of their ages is the number of the house across the street (he points towards the house across the street). What is each daughters' age?" The second mathematician says "it is impossible!" At this point, the first mathematician then relents, providing the final piece of information: "my eldest daughter loves chocolate." The second mathematician now says "Ok, now I know each of your three daughter's ages."
What are the three daughter's ages?
This is not an original problem. Credit goes to Youtube's ScamSchool.
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1 solution
Let the three daughters' ages be represented by x, y, and z. We know xyz = 72. So, (x,y,z) is one of {(1,1,72), (1,2,36), (1,3,24), (1,4,18), (1,6,12), (1,8,9), (2,2,18), (2,3,12), (2,4,9), (2,6,6), (3,3,8), (3,4,6)} Now, the second piece of information is especially tricky. We do not know what the number of the house across the street is, BUT the key insight is this: The two mathematicians know. This means, that for the second mathematician to say it is impossible at this point, the sum x+y+z of at least two elements of the above set must be equal, because if they were all different, then the second mathematician would be easily able to identify the correct element from the set by matching the sum with the number of the house across the street. Mapping the above set to the following set containing the sums of all the elements, we get the following: {74, 39, 28, 23, 19, 18, 22, 17, 15, 14, 14, 13}. Notice that the only two elements with non-unique sums are (2,6,6) and (3,3,8). Now, the final piece of information comes in. This is hidden in the grammar of the last statement: "my eldest DAUGHTER loves chocolate". This implies that there is only one eldest daughter, which means that the largest age must be unique, and we can see that only (3,3,8) satisfies this condition.