# At Tom's Restaurant, Tom will give a free lunch to anyone who can correctly guess the ages of his three children. Joan is a customer who is ready for the challenge. Tom says "First of all, the product of their ages is 36." Joan thinks for a moment then says "That's not enough information. "Okay, I'll give you another hint. The sum of their ages is the same as the number above the restaurant door," says Tom. Joan looks above the door, then writes some notes on a napkin then says "That's still not enough information!" Finally, Tom says "Oh, my oldest child loves banana bread!" After a few seconds, Joan knows the ages of Tom's three children. What are the ages of Tom's three children and did she figure it out?

The first clue tells us that: a X b X c = 36 36 ÷ 1 = 36 36 ÷ 2 = 18 36 ÷ 3 = 12 36 ÷ 4 = 9 36 ÷ 6 = 6 The second clue tells us that: a + b + c = x However, it also tells us that the answer isn't obvious from that second clue. That rules out any of five results that are after the '=', because they are obviously factors of 36 that can be multiplied by one other number, and 1. This completely rules out 36 ÷ 1. That is because 38 is obviously 36 + 1 + 1. The third clue tells us that: a ≠ c a ≠ b That rules out 36 ÷ 6, because the two oldest are the same. Now we are down to: 9, 2, 2 6, 3, 2 4, 3, 3 9 + 2 + 2 = 13 6 + 3 + 2 = 11 4 + 3 + 3 = 10 It is possible that the younger two cannot eat banana bread. That would mean that they would be younger than 3. That means that the ages are 9, 2 and 2.