### Brain Teasers

# Penultimate Frisbee

The game of penultimate frisbee has two scoring moves: the bungee and the wedgie.

Each type of score is worth an odd prime number of points, with a bungee being worth more than a wedgie. If the largest score that can never be achieved in a game is 95 points (no matter how long the game lasts), how many points is each move worth?

Each type of score is worth an odd prime number of points, with a bungee being worth more than a wedgie. If the largest score that can never be achieved in a game is 95 points (no matter how long the game lasts), how many points is each move worth?

### Hint

If the two moves are worth p and q, the largest score that can not beachieved by some combination of bungees and wedgies is p*q-p-q.

### Answer

The bungee is worth 17 points, and the wedgie is worth 7.The largest score that can not be achieved by some combination of p and q points, with p relatively prime to q, is p*q-p-q = (p-1)*(q-1)-1.

So 96 must be factorable as (p-1)*(q-1), for odd primes p and q.

Since the prime factorization of 96 = 2^5*3, and each of the two

factors must be even, the possible factorizations are:

2*48

4*24

8*12

16*6

Only the last factorization yields two factors each of which is one less

than an odd prime, so 17 and 7 must be the two primes we seek.

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