## Circles in a circle

Alan and Bryan meet one Sunday morning to play a game. They sit together at a perfectly circular table with a bag full of perfectly circular coins. The coins are identical. The diameter of the coin is less than that of the table. There is no hole in the table. They have enough coins to cover the entire table. Each player takes turns placing one coin on the table such that no coin touches any other coin or the edge of the table. The game ends when nobody can place a coin. The last one to place a coin wins.

If Alan takes the first turn, who can definitely win the game?

[SOLVED]

#### Alex Yeilding solved this puzzle:

Alan wins. He puts his coin in the center of the table, and at each subsequent move, places his coin exactly opposite Bryan's (reflected in the center of the circle).

#### Vikram Agrawal solved this puzzle:

If Alan is taking the first turn then he can win definitely. He has to put the first coin exactly at the center. After that just put coins exactly diameterically opposite to Bryan's coins.

Since table is circular and so are the coins, it is gauranteed that whenever Bryan puts a coin there will be space available on the diameterically opposite side due to symmetry. So, as long as Bryan can put a coin Alan can also put one. Hence, this way Alan will put the last coin if he follows this strategy.

#### Adam Callanan solved this puzzle:

If Alan has the first move, Alan can win the game every time.

All that Alan has to do to guarantee victory is to place his first coin in the exact center of the table, and then mirror Bryan's every move. Alan's first move is in the center, which Bryan can't mimic, since there is only one center. After that, Bryan can place a coin in any arbitrary spot. There will always be a second arbitrary spot equidistant from the center so that the three coins form a line. Since Alan will have a guaranteed spot mirroring Bryan's coin, eventually Alan will take the last possible spot, by mirroring Bryan's final move.

#### Ian Robertson solved this puzzle:

Alan will win the game.

Alan's winning strategy consists of placing a coin in the center of the table. Alan then mirrors every move of Bryan', i.e. he places his coin so that his original coin is at the midpoint of the line connecting his move and Bryan's move). This ensures that if Bryan has a move so will Alan.

### Credit

This puzzle is taken from folklore.