*(a summary of play-tester feedback; rules available here)*

Your collective impressions of this game squared almost precisely with mine:

- Its simplicity makes it quick and easy to learn.
- As a mathematical puzzle, it’s rich and rewarding.
- As an actual game, it’s a trifle dry.

A typical experience:

I sent this to my group of friends last Sunday night, and by the next morning, two of them had independently produced winning strategies for Player 1, and no one wanted to play it because it had already been solved (sorry).

In my book, I plan to treat each game as a window into mathematical thinking. Our susceptibility to framing effects; the combinatorial nature of creativity; and so on. But Domineering doesn’t necessarily invite grand lessons about the nature of thought. It’s more of a good puzzle.

Thus, I plan to include it in a potpourri-like section called “Puzzle Games,” offering a rat-a-tat collection of simpler fare (with less pontificating from me). Others will probably include Sid Sackson’s Hold That Line and Walter Joris’s Black Hole, among others.

That said, Domineering does lend itself to some gorgeous mathematical analysis. That’s because it quickly breaks up into a collection of smaller games.

Now, this isn’t necessarily a great feature as a *game*:

Would be more interesting to me if it didn’t split into independent subgames so easily, or if those subgames interacted weakly.

But it’s a brilliant feature as a *puzzle*. If you want to understand Domineering as a whole, you simply need to understand the smaller boards that comprise it, and build from there.

For example: no matter whose turn it is, Red wins this board by 1 move. Well call that **+1**.

By the same token, this is a win-by-one-move scenario for Blue. We’ll call that **-1**.

What if we put the two together? That is, what if the board has two regions remaining: a vertical strip of two, and a horizontal strip of two?

Then the winner is whoever plays second. Since it’s a kind of “neutral” board, the result of adding **+1** to **-1**, we’ll call this **0.**

What about, say, this L shape?

If Red plays first, then Blue still has a safe move, and wins. Meanwhile, if Blue plays first, then Blue can make it impossible for Red to move, and thus, Blue wins again.

In short: it’s a winning board for Blue. This suggests it should have a negative value. And yet, check this out: if we combine it with +1 (i.e., a vertical strip), then the outcome becomes a winning board for Red.

What, then, is the value of this L-shaped region?

Here’s the key insight: if you combine two L’s with one vertical strip, the result is a “neutral” board, in which the winner is whoever plays second.

In other words, calling the region’s value L, we have the equation **L + L + 1 = 0**. This has only one solution: **L must be precisely -1/2.**

In other words, this L-shaped region is worth “half” a move for Blue!

This is just a thimbleful of snow scooped from the tip of the iceberg. We haven’t touched on scenarios where the *first* player always wins (e.g., a 2-by-2 square), and what “number” we’d assign to those. (Hint: not a number you’d recognize as a number.) And we’ve barely glimpsed the power of this whole numbering system, which applies to an extraordinary variety of games, and which John Conway considered his finest contribution to mathematics.

My book will only gesture at these depths. But I do hope to give a taste of combinatorial game theory. It’s zesty stuff.