# Space race

This Riddler puzzle is about a game involving filling up the space on a square table using coins.

Two players are seated at a square table. The first player places a coin on the table, the second places a coin on the table, and they carry on placing coins one after another, with the only condition being that the coins are not allowed to touch. The winner is the person who places the final coin on the table, meaning that he or she fills the last remaining space between the other coins.

The table has to be larger than a single coin, and all the coins placed must be identically sized. If the players play optimally, is one of the two players guaranteed to win? If so, what is the winning strategy?

Need a hint?
[Show Solution]

Here is my solution:
[Show Solution]

## 8 thoughts on “Space race”

1. This result also holds for tables that are not square, it only has to be symmetrical under rotation of 180 degrees about the center of table.

Your proof also holds for dimensions higher than 2, for example placing spheres inside a cuboid.

2. Mike says:

The shape can be symmetrical under rotation of 180 degrees (point symmetric) so long is there’s a place at the center (or anywhere else) to have one coin (otherwise it can have holes wherever).

I think it might also permit quite a bit of skewing and have the strategy still work, but haven’t analyzed this to be sure.

1. Shiang Yong says:

Good point Mike. Another way to think of this is the first coin must not break the symmetry. If there is a hole then I think it can be shown that the second player will always win.

3. Dmitry Litvinenko says:

The (pseudo-)solution here is completely unsatisfactory.
The original formulation of the problem asserts only the identity of coins,
but says nothing about the internal symmetry of each individual coin.

1. Not sure what you mean by “internal symmetry”. I did assume that the coins are circular, which I think is a reasonable thing to assume about a coin… But my solution also works for non-circular coins, as long as the coins have the same rotational symmetry mentioned in the other comments, e.g. ellipses, regular polygons with an even number of sides, etc.

Feel free to suggest an alternative solution!

1. Hector Pefo says:

Rotational symmetry of coins is not necessary. As long as they are identical they can be placed with rotational symmetry from the center. Right?

1. That’s true for all the coins except the first one. The first coin needs rotational symmetry!

1. Hector Pefo says:

Right, sorry!