Some Thoughts on Solomon's Stones

Solomon's Stone's is a simple abstract game from Solbenk. They sent me a copy to evaluate and review, which I did. My take was that it looked like a nice game for casual players and non-gamers, but held more interest as a puzzle for gamers.

Rules Review

On your turn, take any number of stones from a single row or column. Last person to take a stone loses.

First Analysis

Our first idea was to look at smaller triangles of size N. I.e.

O

(N = 1)

O
OO

(N = 2)

O
OO
OOO

(N = 3)

and so on, trying to solve the simpler patterns. Clearly, if a game of size N is a win for the second player, then a game of size N+1 is a win for first player, who removes the largest row or column, thereby reducing the game to N.

Game N = 1 is a win for second player.
Game N = 2 is a win for first player, who reduces the game to N = 1.
Game N = 3 is a win for first player, who takes stones A1 + C1 (top and bottom left on above diagram).

This didn't seem to be forming any patterns; N = 4 was not immediately obvious. One of my group members claims to have solved N = 4 and N = 5 as first player wins, but I didn't hear any more on the subject. That was a month ago.

The publisher wrote to me that some people tried to solve the game using symmetrical moves, but that this wasn't ultimately successful.

Second Analysis

I thought a bit more about it this weekend.

Let's start with some new terminology. A(N) is an arrangement of N pieces on a Solomon's Stone board such that all N pieces are on different rows and columns. E.g. for A(7) there is only a single possible arrangement:

O
XO
XXO
XXXO
XXXXO
XXXXXO
XXXXXXO

The stone in row 1 must be in column 1. The stone in row 2 must be in any column that the stone in row 1 doesn't occupy, which must be column 2. And so on.

For A(6), there are 127 possible arrangements:

I. There is one solution for A(6) if the stones occupy the first six rows, and it looks similar to the above diagram.

II. There are two solutions for A(6) if the stones occupy rows 1 through 5 and row 7. Row 7's stone can be placed either in column 6 or column 7.

III. There are four possible solutions for A(6) if the stones occupy rows 1 through 4 and rows 6 and 7. If row 6's stone is place in column 5, row 7's stone can be placed in column 6 or 7. If row 6's stone is placed in column 6, row 7's stone can be placed in column 5 or 7.

The pattern is now obvious. For the shape:

XO
XXO
XXXO
...

The first row has two possible choices. For either choice, the next row has two possible choices. And so on, leading to 2^N possible arrangements.

Continuing from above,

1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 (or simply 2^(N+1)-1)

First player loses for A(odd), while second player loses for A(even). One winning strategy is therefore to force your opponent to play an A(odd) position. If you see a position which can be reduced to A(odd), you win. Let's call this position R(1). R(1) is a position that can be reduced in one play to A(odd). All A(even) positions are R(1).

You should not leave your opponent in R(1). In fact, you want to try to force your opponent to leave you in R(1).

Unfortunately, I believe that a savvy opponent should always be able to, rather than leave you in R(1), remove a necessary stone for your A(odd) configuration. Seeing as there are 127 possible A(6) configurations, not to mention A(4) and A(2) configurations, you may be able to set up a fork situation, so that your opponent cannot block both A(even) positions at the same time.

At least, this is where I got to in my thinking over the weekend. With a board as small as Solomon's Stones, it's possible that I'm overlooking a simpler solution.

Yehuda

Review: Solomon's Stones

Solbenk is a new subsidiary company with two new games: Solomon's Stones and Saikoro (the latter covered in another post). The publisher offered to send me copies of the games to review.

Solomon's Stones, pictured above, is a simple abstract based on a classic mathematical concept called "nim". The game adds a bit of a twist to the classic nim game, however.

Rules

The game is for two players an plays in three to twenty minutes, depending on how much time is spent thinking about the moves.

There are seven rows and seven columns of stones, twenty-eight stones in all arranged in a triangle (see above picture).

On your turn, you must remove one or more stones from a single row or column. The removed stones do not have to be adjacent, and you may remove as many or few stones as you like, so long as you remove at least one and all come from the same row or column.

The player who removes the last stone loses.

Components

It's a pretty game. The board is made of black plastic, and the stones are made of hematite gemstones, which feel something like glass. While the stones are pretty enough for a coffee table, a wooden board would have completed the package; the plastic board looks perfectly fine for a game, but not good enough for a coffee table.

There is a half-page insert with the rules.

Reactions

I played one game and it was enjoyable enough. I believe that, like Quarto, Connect Four, and other elegant abstracts, it will serve as a diverting activity for non-gamers. If you like those games, you'll probably like this one, too.

But, like Quarto, Connect Four, and other such games, I doubt very much that I will play it again. Because it works so much better as an excellent mathematical puzzle than as an ok game.

My friends and I sat down and began examining it to see how it could be solved. We looked at the trivial cases, different patterns, mirror moves, and so on. When we're done, we'll post our analysis. In the meantime, although the game seems quite stark and simple, it's entirely possible that it's NP-complete.

I expect to take some time over the next week or two thinking about how to solve it.

Tagline

The tagline of the game is "an original classic strategy game", which is oxymoronic.