[email protected] wrote:Coins are to be put into 7 pockets so that each pocket contains at least one coin. At most 3 of the pockets are to contain the same number of coins, and no two of the remaining pockets are to contain an equal number of coins. What is the least possible number of coins needed for the pockets?

We want to minimize the number of coins. Since there must be at least 1 coin per pocket, let's put 1 coin in the first 3 pockets.
Pocket #1: 1 coin
Pocket #2: 1 coin
Pocket #3: 1 coin

No two of the remaining pockets are to contain an equal number of coins.
So, from this point on, we cannot put 1 coin in any more pockets (otherwise, we'll break the rule about no more than 3 pockets containing the same number of coins).
To minimize the number of coins, we'll put 2 coins in pocket #4, then 3 in pocket #5 and so on...
Pocket #4: 2 coins
Pocket #5: 3 coins
Pocket #6: 4 coins
Pocket #7: 5 coins

So, the minimum number of coins = 1+1+1+2+3+4+5 = 17

Cheers,
Brent