Three
Variations on a Standing Bosses Theme
1. Consider the following
variation on the "Standing Bosses" game:
Line up some players, sitting in chairs, facing the
class, then iterate:
- The boss is the second-to-rightmost sitting player
- The boss’s job is to stand up, and to tell all standing players
to his right to sit down
The game is repeated for as long as possible.
For example, with 3 players, the game goes like: 000
-> 010 -> 100 -> 110 and game over.
- The game with 3 players had 4 positions altogether. How
many positions does the game with 4 players have altogether?
- How many positions do the games with 1, 2, and 5 players have?
- How many positions does the game with n players have?
2. Consider the following variation on the "Standing Bosses" game:
Line up some players, sitting in chairs, facing the
class, then iterate:
- The boss is the rightmost sitting player who does not have a
person standing to his immediate left
- The boss’s job is to stand up, and to tell all standing players
to his right to sit down
The game is repeated for as long as possible
For example, with 3 players, the game goes like: 000
-> 001 -> 010 -> 100 -> 101 and game over.
- The game with 3 players had 5 positions altogether. How
many positions does the game with 4 players have altogether?
- How many positions do the games with 1, 2, and 5 players have?
- How many positions does the game with 15 players have?
3. Consider the following variation on the "Standing Bosses" game:
Line up some players, sitting in chairs, facing the
class, then iterate:
- The boss is the person to the left of the leftmost sitting
standing
player, unless the leftmost sitting
standing player
is the leftmost player, in
which case the boss is the rightmost player.
- The boss’s job is to stand up, and to tell all standing players
to sit down
The game is repeated for as long as possible.
How many positions does the game with n players have?