ekpocket.html

Coins in my pocket

by Gil French

Suppose you have 35 coins and 9 pockets. Is it possible to put a different number of coins in each pocket? (Having 0 coins in a pocket is allowed.)
In each case below, find the minimum number of coins you can have, if you have a different number of coins in each pocket.
1. You have 3 pockets.
2. You have 4 pockets.
3. You have 5 pockets.
Suppose you have n pockets. What’s the minimum number of coins you can have, if you have a different number of coins in each pocket?

It’s not possible. If you assume the least number in the first pocket--0 coins--the next least (1 coin) in the second pocket, and so on, you get 36 coins for 9 pockets. For 35 coins, you could have 8 pockets with 0, 1, 2, . . . , 7, but the ninth pocket could only have 7 coins.
Using the same kind of reasoning:
1. at least 3 coins (0+1+2)
2. at least 6 coins (0+1+2+3)
3. at least 10 coins (0+1+2+3+4)
The number of different coins is the sum of the first n - 1 counting numbers. (Since 0 doesn’t contribute to the sum, there are only n - 1 numbers you have to add together.) This is also .