Fewest Tokens: Pay 6
Read the puzzle
You must pay exactly 6 using tokens worth 1, 3 and 4. You may use as many of each token as you like, and no change is given. Which tokens pay 6 using the fewest tokens possible?
Which tokens pay 6 with the fewest tokens?
Biggest first is not always fewest. Try every pair.
3 + 3
Just 2 tokens. Biggest first, 4 + 1 + 1, needs 3.
How to solve it
Search by count, not by size: ask whether 1 token works, then 2, then 3.
One token: no token is worth 6, so one is impossible.
Two tokens: try every pair. 1 + 1 = 2, 1 + 3 = 4, 1 + 4 = 5, 3 + 3 = 6, 3 + 4 = 7, 4 + 4 = 8. Only 3 + 3 hits 6 exactly. So the fewest is 2 tokens: 3 + 3.
The usual wrong answer is 4 + 1 + 1. It comes from the "biggest first" habit: take the largest token that fits (4), leaving 2, then fill the 2 with two 1s. That pays 6 but uses 3 tokens, one more than needed. Grabbing the 4 first locks you out of the better pair.
Takeaway: with odd token sets, greedy choices can lose. Checking every pair is quick and settles it.
Learn the method: Coin Total Puzzles: Largest Unpayable Total and Fewest Tokens
