# Thinking between the Lines | Solution

Jack says, “They’re 3, 3, and 8.”

How did Jack arrive at that? Let’s look at what was going on in his head before he laughed and told Bill he couldn’t solve the puzzle.

The first thing Jack did was figure out all the possible ways to factor 72.

First, he divided 72 by 2 until the quotient is odd:

72 / 2 = 36

36 / 2 = 18

18 / 2 = 9

And he knows that 9 is a perfect square, i.e. 9 = 3×3. So, he knows then that 72 = 2x2x2x3x3.

He also knows that 1 multiplied by anything leaves it unchanged, so age 1 is also a possibility: 72 = 1x2x2x2x3x3

The ages he can play around with, then, are all the ways he can group the product of numbers into threes:

(1) x (2) x (2x2x3x3) → Ages 1, 2, 36 (sum = 1 + 2 + 36 = 39)

(1) x (2×2) x (2x3x3) → Ages 1, 4, 18 (sum = 1 + 4 + 18 = 23)

(1) x (2×3) x (2x2x3) → Ages 1, 6, 12 (sum = 1 + 6 + 12 = 19)

(1) x (2x2x2) x (3×3) → Ages 1, 8, 9 (sum = 1 + 8 + 9 = 18)

(1) x (2x2x2x3) x (3) → Ages 1, 3, 24 (sum = 1 + 3 + 24 = 28)

(1×2) x (2) x (2x3x3) → Ages 2, 2, 18 (sum = 2 + 2 + 18 = 22)

(1×2) x (2×2) x (3×3) → Ages 2, 4, 9 (sum = 2 + 4 + 9 = 15)

(1×2) x (2×3) x (2×3) → Ages 2, 6, 6 (sum = 2 + 6 + 6 = 14)

(1×2) x (2x2x3) x (3) → Ages 2, 3, 12 (sum = 2 + 3 + 12 = 17)

(1x2x2) x (2×3) x (3) → Ages 3, 4, 6 (sum = 3 + 4 + 6 = 13)

(1x2x2x2) x (3) x (3) → Ages 3, 3, 8 (sum = 3 + 3 + 8 = 14)

Now, when Jack did this all in his head, the reason he laughed and said to Bill, “You expect me to figure that out?” wasn’t because it was hard. It was because, as you can see, he realized that not every sum was unique.

In fact, of all the sums, only two are the same: 14, which you get from ages 2, 6, and 6, and from ages 3, 3, and 8. Jack laughed at Bill because Bill did not give him a fair puzzle.

Certainly, if Jack had calculated everything and found one sum that matched his birthday, he would have been able to figure it out. The fact that he didn’t implies Jack’s birthday is the number repeated twice (14). And that’s the twist in this one.

When Bill told Jack that his eldest child just started piano lessons, Jack replied without asking further questions.

This is because at this point, Jack is only concerned with the two possibilities: ages 2, 6 and 6, and ages 3, 3, and 8. But if they are 2, 6, and 6, then there is no eldest. Therefore, it has to be the other possibility: ages 3, 3, and 8.

You can see here that the real trick to this kind of puzzle is thinking between the lines. Why did Jack really laugh, and what was he thinking? Just like that, we’ve also figured out his birthday.

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