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The teacher's daughters are 2, 2 and 9 years old.

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Solution by Jason Meyers

There are 8 combinations of 3 numbers that equal 36 when multiplied, and their sums are in parenthesis:

1, 6, and 6 (13)
1, 4, and 9 (14)
2, 3, and 6 (11)
2, 2, and 9 (13)
3, 3, and 4 (10)
1, 3, and 12 (16)
1, 2, and 18 (21)
1, 1, and 36 (38)

Caesar could easily figure this out, but doesn't know their ages yet. 

When told their sum is the same as the house across the street, Caesar can look at that number and quickly find out which of these combinations is correct, unless the number of the house is 13, for which there are two possible sums. 

Since he cannot solve the problem by knowing the house number, 13 must be the number of the house. 

When told that the eldest daughter plays piano, he knows that it cannot be the 6, 6, and 1 combination because their isn't an eldest as the twins are the same age, while the 2, 2, and 9 combination has an eldest at age 9, so that must be it.

Solution by Jensen Lai

The three ages multiply to give 36. The possible combinations are as follows:

1,1,36 total 38
1,2,18 = 21
1,3,12 = 16
1,4,9 = 14
1,6,6 = 13
2,2,9 = 13
2,3,6 = 11
3,3,4 = 10

Caesar knew the sum of the numbers yet still couldn't figure out the ages. Thus, the house number must have been 13. Knowing that there was an oldest girl, Caesar figured out that the ages were 2,2 and 9, not 1,6 and 6. Therefore, the girls are aged 2,2, and 9.

Solution by Marcus Dunstan

The girls' ages are 9, 2 and 2

Method:
Oldest girl must be either 18,12,9 or 6 (in order that the 3 ages can multiply to make 36)
Knowing the house number, Caesar should be able to determine the right ages UNLESS there are 2 combinations of the 3 ages that add up to the same number. There are:
9,2 and 2 (=13) and
6,6 and 1 (=13)
Because the teacher referred to his oldest daughter (in the singular) then the solution must be 9, 2 and 2!

Solution by Glenton Jelbert

The ages are 2,2 and 9. 

Look at all the possible combinations of 3 positive integers whose product is 36. There are 8. The sum of the 3 numbers for these eight are 38, 21, 16, 14, 13, 13, 11, 10. Since the house number is not sufficient to tell the ages of the daughters the house number must be 13, corresponding to daughters ages of {1, 6, 6} or {2, 2, 9}. Since there is an eldest daughter it must be the later.

Solution by Tina Nolte

The ages of the daughters are: 9, 2, and 2.
The thing to note is that the third statement of the teacher revealed two things: the first two statements underdetermined the answer and knowing there was a unique oldest would allow us to determine which set of ages was correct. Considering the set of natural numbers that would multiply to 36 and would add up to the same number as some other such set of numbers gives us only two (multi)sets: (2, 2, 9) and (1, 6, 6). However, the statement about the oldest playing the piano tells us that there _is_ an oldest, so it must be the multiset (2, 2, 9).

Solution by Kimberly Puen

The girls are ages 2, 2, and 9
2x2x9=36 
2+2+9=13 (similar to 1+6+6=13, but there's an oldest girl)

Last Updated: January 15, 2007  |  Posted: March 19, 2002
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