Two teams played a game called Thai 21 on an episode of the television show Survivor. Call the teams Green and Red. The game begins with 21 flags. The teams take turns. When it is a team's turn, it can remove one, two, or three flags
The team that removes the last flag wins. Green goes first. Who should win this game, Green or Red?
Green. Use backward induction to solve this game. If it is Green's turn and there are one, two, or three flags left then Green can take the remaining flags and win. If there are four left and it is Red's turn then Green will win. Whatever Red does it will be Green's turn and one, two, or three flags will remain; we have already seen that Green will win in that case. If it is Green's turn and there are five, six, or seven flags left then Green will win. Green takes the number of flags so that four flags remain (so, for example, if seven flags remain then Green takes three). We have already seen that if four flags remain and it is Red's turn then Green will win. If you continue this line of logic you should see that Green will win this game by playing the following strategy. Take one flag at the start of the game so that 20 remain. Then take enough flags so that when it is Red's turn 16, 12, eight, and four remain. Then take the remaining flags and win.
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