Challenge 1
Imagine you have an endless supply of 5p and 7p coins. You could make exactly 20p with four 5p coins. Or you could make exactly 19p with two 7p coins and a 5p coin.
What is the biggest amount that you cannot make? Can you explain why you can make all amounts after this one?
Challenge 2
The biggest number you cannot make given coins of value x and y is called the Frobenius number.
Can you think of a pair of numbers that wouldn’t have a Frobenius number?
What must be true about the two numbers for them to have a Frobenius number?
Challenge 3
Can you come up with a formula for the Frobenius number, if it exists, for two numbers x and y?
Solutions here.