A friend of mine, Sol announces that he has a number of coins in his pocket that add up to precisely one dollar. When he tells me how many coins he has, I ask if any one of them is a half-dollar. Sol tells me "no". I realize that I can't tell for sure what coins he has, because there are six different combinations that produce precisely one dollar.
How many coins does Sol have in his pocket?
Each of the six possibilities of Sol's coins can be transformed to the others through one or more of the following relationships:
1 quarter + 3 nickels = 4 dimes
4 dimes + 5 pennies = 9 nickels
1 quarter + 5 pennies = 6 nickels
3 quarters + 5 pennies = 8 dimes
Each relation preserves the total value and total quantity of coins.
From here, I just played with possible starting configurations until I got a solution with 16 coins:
2 quarters, 4 dimes, 10 pennies
2 quarters, 9 nickels, 5 pennies
1 quarter, 4 dimes, 6 nickels, 5 pennies
1 quarter, 15 nickels
8 dimes, 3 nickels, 5 pennies
4 dimes, 12 nickels