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Problem of the Week
Problem C and Solution
The Snowflake Game

Problem

The Snowflake Game is played on a board that consists \(9\) circles. There is \(1\) circle in the centre of the board, with \(4\) line segments through it, and a circle at the end of each line segment.

Two players alternate turns placing discs numbered \(1\) to \(9\) in the circles on the board. Each number can only be used once in any game. The object of the game is to be the first player to place a disc so that the sum of the three numbers along a line segment through the centre circle is exactly \(15.\)

Alex and Blake play the game. Alex goes first. Show that Alex has a winning strategy if she places a \(6\) in the centre circle on her first turn. That is, show that if Alex places a \(6\) in the centre circle on her first turn, then no matter which numbers Blake plays, Alex can always win the game.

Solution

If Alex places a \(6\) in the centre circle on her first turn, then the other two discs in the line segment would need to add to \(9\) in order for the sum to be \(15.\)

Since each number can be used only once in a game, after Alex places the \(6\), the numbers that remain are \(1\), \(2\), \(3\), \(4\), \(5\), \(7\), \(8\), and \(9.\)

Therefore, we have shown that if Alex places a \(6\) in the centre circle on her first turn, then no matter which numbers Blake plays, Alex can always win the game on either her second or third turn.

For Further Thought:

What other numbers can Alex place in the middle circle on her first turn to have a winning strategy? That is, what other numbers can Alex place in the middle circle on her first turn so that she can always win the game, no matter which numbers Blake plays?