How can you guarantee victory in this game?

Question

15 pennies, 2 players. Each player removes 1-5 pennies per turn. The player to remove the last penny wins. How can you guarantee victory?

I know this has something to do with leaving the other player with 6 pennies on their last turn. But what comes before that?

Answer 1

assume A & B are the players

the condition is to remove 1-5 pennies at a time

suppose A plays first - for him to win, he must play 3 leaving 12; then he plays 6-b, b being whatever B plays

now let us find B's his winning strategy

work backwards....as given

B must leave 6 in his penultimate round; so 9 must be played with his turn

the strategy is to remove 9-x where x is what A played, assuming x>=4

if x<4, then B must play 6-x....

Answer 2

On the first turn, player #1 takes 3, leaving 12 pennies. This ensures he will leave 6 pennies after player #2 makes his move.

Answer 3

Just think backward. What previous move would guarantee that you could leave 6 pennies?

Answer 4

If you can play first, remove three pennies.
After that, remove 6 minus the number your opponent played.

Or, even more generally: always remove as many as are needed to have a multiple of 6 left.