Arithmetic+ 07: UNIZOR.COM - Math+ & Problems - Arithmetic

Notes to a video lecture on http://www.unizor.com

Arithmetic+ 07

Problem A

Two friends mathematicians, Mike and David, met after a long time and Mike tells that he has three sons.
"How old are they?" - asked David.
"Sum of their ages is 13." - answered Mike.
The David replied: "I see, you are challenging me. What else can you tell, so I can determine their ages?"
"The product of their ages is the number of windows in that building across the street." - said Mike.
David counted the windows in the building and said: "But this is still insufficient to determine the ages of your sons."
Mike added then: "Take into consideration that my oldest son is a redhead."
David responded: "Then I can say that their ages are..."
"That is correct." - said Mike.
What are the ages of Mike's sons?

Solution A
There are many combinations of three natural numbers with a sum of them equal to 13.
We can write down all of them.
For each such combination we can calculate the product of its numbers.
1 × 1 × 11 = 11,
1 × 2 × 10 = 20,
1 × 3 × 9 = 27,
1 × 4 × 8 = 32,
1 × 5 × 7 = 35,
1 × 6 × 6 = 36,
2 × 2 × 9 = 36,
2 × 3 × 8 = 48,
2 × 4 × 7 = 56,
2 × 5 × 6 = 60,
3 × 3 × 7 = 63,
3 × 4 × 6 = 72,
3 × 5 × 5 = 75,
4 × 4 × 5 = 80
If all products are different, that would be sufficient for David to determine which combination represents the ages of Mike's sons, because he would compare the products with the number of windows he counted.
However, since David said that the information about the product is insufficient, it means that there are some combinations that give the same product.
Only 36 is repeated for two combinations, so the answer must be among them.
Information that the oldest son is a redhead resolves the dilemma in favor of (2,2,9), because (1,6,6) combination has two oldest sons.

Answer A
The ages of three Mike's sons are 2, 2 and 9.