Removing Flags (Strategy)

[BigBeachBanana]

Suppose there are 21 flags. Team A and B take turns to remove the flags. Team A always goes first. Each team must remove 1,2, or 3 flags per turn. The team that removes the last flag wins. One team will always win the game.
Questions: Which team is it? What's that team's strategy?

 

[Jones]

Team A will always win.

Whoever grabs the last flag wins, and you need to be able to get there within 3 steps. If Team A lands on the 5th to last flag, then Team B will be just out of reach of the last flag, and if they move even one step, Team A will be able to win. Effectively, whoever lands on the 5th to last flag wins. We can continue this logic backwards, where each flag 4 spots down is a "victory" flag.

The first victory flag is the very first flag in the sequence, and since Team A always goes first, they'll be able to grab that flag. They must only grab that flag, however, otherwise Team B will be able to pick up the next victory flag. To secure each victory flag from there, Team A must choose a number of flags such that the total flags removed within one turn cycle is 4. So if Team B picks 1, 2, or 3, Team A picks 3, 2, or 1 respectively.

Would be curious to know how this problem might be solved more analytically.

 

[Steven G]

The way I played this game is as followers. Play starts with the number = 0. Each player takes turns saying numbers but the number they say must be 1, 2 or 3 more than the last number.

 

[Steven G]

Would be curious to know how this problem might be solved more analytically.

Work backwards.
Player 1 takes flag number 21
What number flag(s) can player 1 takes before her last turn.
...

 

[lev888]

4n+1 flags on your turn always leads to a loss - can be proven by math induction. B wins by leaving 4n+1 flags.