In my left pocket I have a mix of pennies (1 cent) and quarters (25 cents). In my right pocket I have a bunch of dimes (10 cents). The number of coins in each pocket is the same, so is the cash value. What is the smallest (nonzero) number of coins I can have?

What if I had coins of x cents and z cents in my left pocket and coins of y cents in my right pocket - is there some quantitiy of coins I can have so that each pocket has the same number of coins and same cash value? (The value of the coins are positive integers x > y > z.)

Part I
16 coins:
3 quarters + 5 pennies = 8 dimes

Part II
(Assuming coins involved are limited to those in current circulation of U.S. currency...) only multiples of 6, 8, 22, 38, and 98 coins can result in a quantity of coins in each pocket with the same cash value. 

6 coins:
1 dollar + 2 quarters = 3 half-dollars

8 coins:
1 quarter + 3 nickles = 4 dimes

12 coins:
[6 coins] x 2
1 quarter + 5 dimes = 6 nickles
1 dollar + 5 dimes = 6 quarters

16 coins:
[8 coins] x 2
3 quarters + 5 pennies = 8 dimes
3 half-dollars + 5 dimes = 8 quarters

18 coins:
[6 coins] x 3
4 dimes + 5 pennies = 9 nickles
1 half-dollar + 8 nickles = 9 dimes
4 half-dollars + 5 nickles = 9 quarters
4 dollars + 5 dimes = 9 half-dollars

22 coins:
1 dollar + 10 pennies = 11 dimes

38 coins:
1 dollar + 18 nickles = 19 dimes
4 dollars + 15 nickles = 19 quarters
9 dollars + 10 nickles = 19 half-dollars

66 coins:
[22 coins] x 3
8 dollars + 25 pennies = 33 quarters

98 coins:
4 half-dollars + 45 pennies = 49 nickles
9 half-dollars + 40 pennies = 49 dimes
24 half-dollars + 25 pennies = 49 quarters

198 coins:
[6 coins] x 33
[18 coins] x 11
[22 coins] x 9
[66 coins] x 3
4 dollars + 95 pennies = 99 nickles
49 dollars + 50 pennies = 99 half-dollars

Edited on February 17, 2009, 2:25 pm

Posted by Dej Mar on 2009-02-17 14:15:11