## Saturday, May 2, 2009

### Puzzle: Circular Table, Pile of Quarters and 2 Players

Puzzle: There is a huge pile of quarters and a circular table. There are two people to play a game of placing the quarters down on the table alternately without any overlap. The one who can't put down a quarter loses. Assuming that the pile of quarters is non-exhaustive what will be your winning strategy? Whether you would like to start or let your partner start?

Solution: The winning strategy here will be to find a way where you always end up getting a
section of free space on the table to place down the quarter in your hand. That's fine, but how can one always ensure free space on the table? The table is circular and hence due to symmetry every place on the table (except for the centre) will have its own corresponding counterpart right opposite to it on the other side of the centre.

Got the answer now? Since the centre of the table is the only exception (not having a
symmetrically opposite area) here and any other area on the table can always be guaranteed to have a directly opposite area where the centre of the table will be right in the middle of the two.

For winning the game you got to start first and place the quarter exactly in the centre of
the table. From now on, any area your parter chooses to put down the quarter in their hand, will have its corresponding symmetrically opposite area on the other side of the centre of the table where you can place the quarter on your turn.

This will guarantee that you never fall short of free space on the table as this way you would be occupying the last usable (free) section of the table and consequently your partner will be the one failing to find a free area on the table to place the quarter on. How long/soon that happens, will depend upon the size of the quarters and that of the table.

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