Comments on: Bar game: place the last coaster

By: bill

bill — Wed, 17 Mar 2010 21:05:34 +0000

This assumes there’s always a matching spot. Holes in the table, or an irregular outline, would make that difficult. From the comments, I gather that once you no longer mirror, the strategy isn’t assured.

By: Craig

Craig — Wed, 17 Mar 2010 14:46:32 +0000

Presh:
Your solution is an ‘involution’. I actually worked on a combinatorial game similar to this recently. It’s awesome to see this version of it! What a great way to introduce laypeople to game theory, and especially to show them how combinatorialists arrive at solutions.

If you’re interested in another form of the game check out F-Saturator, a game a few colleagues of mine worked on in the form of graphs.

http://scholar.google.com/scholar?cluster=5600507325285475104&hl=en&as_sdt=4000

By: Scott

Scott — Wed, 17 Mar 2010 13:20:35 +0000

One thing that’s unclear is whether or not you can move coasters that have already been placed.

I like the solution and how it works:

Assuming optimal placement (each coaster touches another coaster) we would end up with a rectangular array of coasters. Since you placed an “anchor” at the center of the table, the array of coasters will have an odd number of rows and columns. Since an odd number times an odd number is also an odd number, there would be an odd number of rectangles. Since you went first, you will go last due to that odd number.

By: Presh Talwalkar

Presh Talwalkar — Wed, 17 Mar 2010 06:39:58 +0000

There is a winning strategy for the first player in a 2-person contest. Here is what to do:

Your first move is to place a coaster in the center of the table. Now, wherever your opponent places the coaster, you place yours symmetrically on the other side of the table. If they place a coaster in the southwest corner, you place yours in the analogous spot in the northeast corner. (If you imagine the center of the table as the origin, this is mathematically a reflection about the origin).

This strategy means you can always match your opponent’s move. The game ends when your opponent runs out of open sports, which equivalently means you have placed the last coaster.