Henry Ernest Dudeney/536 Puzzles & Curious Problems/3 - Loose Cash/Solution

by : $3$

 * Loose Cash
 * What is the largest sum of money -- all in current silver coins and no silver dollars --
 * that I could have in my pocket without being able to give change for a dollar, half dollar, quarter, dime or nickel?

Solution
The largest sum possible is $\$ 1.19$, composed of:
 * $1$ half dollar
 * $1$ quarter
 * $4$ dimes
 * $4$ cents.

Proof
The coins available are:
 * The half dollar, which is $50$ cents
 * The quarter, which is $25$ cents
 * The dime, which is $10$ cents
 * The nickel, which is $5$ cents
 * The cent itself.

We can have $4$ cents, but not $5$ cents as then we will have change for a nickel.

We can have $1$ nickel, but not $2$ nickels as then we will have change for a dime.

We can have $4$ dimes, but not $5$ dimes as then we will have change for a half dollar.

But if we have $2$ dimes as well as a nickel, then we will have change for a quarter.

Now $4$ dimes is more than $2$ dimes and a nickel, so for our maximum we need to lose the nickel.

We can have $1$ quarter, but not $2$ quarters as then we will have change for a half dollar.

We can have $1$ half dollar, but not $2$ half dollar as then we will have change for a dollar.

We have the maximum number of all the coins we are allowed.

Hence the result.