### BULLET BRAIN BUSTER

#### Vanilla Ice

We all know that Vanilla Ice just loves Hot Chocolate and he's just bought himself four bumper bags.

He weighs them two at a time in all possible pairs and finds that his pairs of weights total 6, 8, 10, 12, 14, and 16 Kilos. But how much do they each weigh individually?

###### More Caffeine Bullet Brain Busters

There are two possible answers: Vanilla's boxes of hot chocolate can be 1, 5, 7, and 9 kilos, or they can be 2, 4, 6, and 10 kilos.

Let the boxes be a, b, c, and d, sorted such that a < b < c < d. We can chain inequalities to get a + b < a + c < a + d, b + c < b + d < c + d. Thus, a + b = 6, a + c = 8, b + d = 14, and c + d = 16. But we don't know if a + d = 10 and b + c = 12 or the other way around. This is how we get two solutions. If a + d = 10, we get 1, 5, 7, and 9; if b + c = 10, we get 2, 4, 6, and 10.

Where this problem really gets weird is that the number of solutions depends on the number of boxes. For example, if Vanilla has three boxes and knows the weight of all possible pairs, then there is only one possible solution for the individual boxes. The same is true if he has five boxes.

But now suppose that Vanilla has eight boxes, and the sums of pairs are 8, 10, 12, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 36, 38, and 40. Now what are the individual boxes?

This time, there are three solutions:

• 1, 7, 9, 11, 13, 15, 17, 23

• 2, 6, 8, 10, 14, 16, 18, 22

• 3, 5, 7, 11, 13, 17, 19, 21