Fewest Tokens: Pay 24
Read the puzzle
You must pay exactly 24 using tokens worth 1, 6, 10 and 15. Every token type is unlimited, and no change is given. Which tokens pay 24 using the fewest tokens possible?
Which tokens pay 24 with the fewest tokens?
Biggest first is not always fewest. A token can be used many times.
6 + 6 + 6 + 6
Just 4 tokens. Biggest first, 15 + 6 + 1 + 1 + 1, needs 5.
How to solve it
Work up one count at a time and rule each one out before moving on.
Two tokens: 24 would need 15 + 9, 10 + 14, 6 + 18 or 1 + 23. None of 9, 14, 18 or 23 is a token, so two fails.
Three tokens: with a 15, the other two must make 9, but 6 + 1 = 7 and 6 + 6 = 12 miss. With a 10 and no 15, the other two must make 14; 10 + 4, 6 + 8 and 1 + 13 all fail. With only 6s and 1s, three tokens make at most 18. So three fails.
Four tokens: 6 + 6 + 6 + 6 = 24 works. The minimum is 4 tokens: four 6s.
The usual wrong answer is 15 + 6 + 1 + 1 + 1. Biggest first takes 15, leaving 9; then 6 leaves 3, and three 1s finish it. That is 5 tokens.
Quick check: 4 × 6 = 24. Takeaway: a token may be used many times, so repeating one middle-sized token can beat mixing big and small ones.
Learn the method: Coin Total Puzzles: Largest Unpayable Total and Fewest Tokens
