Henry Ernest Dudeney/536 Puzzles & Curious Problems/3 - Loose Cash/Solution
536 Puzzles & Curious Problems by Henry Ernest Dudeney: $3$
- 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.
$\blacksquare$
Sources
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $3$. Loose Cash