Change for a Dollar
Molly McKee
What is the largest amount of money, obviously in coins each less than one dollar, you can have and not be able to make change for a dollar?
The Problem
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First, what coins are available:
Half Dollar
Evaluate the coins, starting with the largest denomination:
How many half dollars can we have and not equal $1?  1 half dollar
If we have 1 half dollar, how many quarters can we have? 1 quarter
    Because any more quarters, you could give 2 quarters and 1 half dollar
Now we have 1 half dollar and 1 quarter? How many dimes can we have? 4 dimes
Now we have 1 half dollar, 1 quarter, and 4 dimes.
How many nickels can we have? 0 nickels
    With 1 nickel: 1 half dollar, 1 quarter, 2 dimes, and 1 nickel = $1
    With 2 nickels: 1 half dollar, 4 dimes, and 2 nickels = $1
That means we can only have 4 pennies, because 5 pennies equals 1 nickel
1 half dollar + 1 quarter + 4 dimes + 0 nickels + 4 pennies = $1.19
Would the total change if we excluded the half dollar?
There is a $2 bill, would the total change if there was a 2 cent coin?