### One of a continuing series

I once took this course in Mathematical Analysis. The text was called *Advanced Calculus*, but it wasn’t all about differential and integral calculus. It had a bunch of other stuff. I still have the book, but I didn’t need to go back to it to do this week’s Quiz Question. I am posting it from memory.

Look at the diagram above. There is a game that involves a large circle. Maybe the circle is on a board and has a rim around it to keep players from encroaching outside the circle. Two players play, and because this is math and not mechanical engineering, their play is perfect. Play is perfect in that players can place tokens within the circle with ultimate precision. Player A has a sufficient supply of yellow tokens, and so does player B. All tokens are perfect circles (mathematics, remember), and all are the same size. The object of the game is to be the last to place a token within the circle. Once you place a token in the circle it cannot be moved, and of course, tokens are allowed to touch but not to overlap.

And here is this week’s Quiz Question: Is there a winning strategy? If so, who owns the winning strategy, and what is that strategy?

Post your answer as a comment below.

## Update and hint

Nobody has attempted a solution, so here’s a hint: The first player places his token in the center of the circle and now has a winning strategy. What is the winning strategy?

## Update and solution

Greg has provided the correct solution. See his comment below.

The first player should play the very center of the circle. From that point on he should play the radial opposite of the second player’s move. That way the first player will always be able to maintain symmetry. Ultimately the second player will be unable to break the symmetry—unable to play.

There is another winning strategy. What is it?

John, given your hint, this looks like a symmetry problem. If the first player plays in the exact center of the large circle, then plays a small circle exactly opposite their opponents play every time, then the first player should win. The reason is that, since symmetry is maintained throughout the game, if the second player can play there must be an open space opposite that the first player can then play to.

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