With only 5-cent and 7-cent stamps, what's the largest amount you can never make exactly?
You have an unlimited supply of stamps in two denominations that share no common factor, say 5 cents and 7 cents. You can combine any number of each. Some amounts are impossible to make exactly, like 1, 2, 3, 4, 6, 8, 9, 11, 13 or 18 cents, but past a certain point every amount becomes achievable. What's the largest amount that can never be made exactly, and is there a formula for it?
Reveal the answer
For two coprime denominations a and b, the largest impossible amount is always a×b − a − b; for 5 and 7 that's 35 − 5 − 7 = 23 cents. This is the two-variable case of what's now called the Frobenius, or Chicken McNugget, problem, first solved in this general form by mathematician James Joseph Sylvester in 1884; finding the equivalent formula for three or more denominations remains, in general, an unsolved problem.
— James Joseph Sylvester, Mathematical Questions with their Solutions — Educational Times, 1884