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A Number of Games

The Quest Coast

It's playtime! Rita and her classmate Lefty have been playing a lot of games lately. So much, in fact, that they are quite familiar with the Winning Ways of all the games they play.

Game rules:

In the games they play, there are two players who alternate making moves. The first time that a player cannot make a move on their turn, that player loses.

Col: Each player, on their turn, paints one region of the map their color (bLue for Lefty, or Red for Rita). Adjacent regions may not share a color.

Domineering: Lefty moves by placing a vertical $2 \times 1$ domino onto two empty squares of the grid. Rita does the same thing with horizontal $1 \times 2$ dominos.

Hackenbush: Lefty moves by deleting any bLue edge, together with any edges that are no longer connected to the ground. Rita does the same thing with Red edges.

Toads and Frogs: On a $1\times n$ strip of squares are Lefty's toads (facing right) and Rita's frogs (facing left). On a player's turn, one of their animals will move forward one space into an empty square, or hop over one opposing animal into an empty square.

From playing the games so much, they have learned a way to give values to their game positions, and to compare those values against each other. Here are some of their game theories:

If Lefty always has a strategy that will win, no matter who goes first, then the game has a positive value. For example, in this Domineering game, Lefty can make one move but Rita cannot make any. It has a positive value. Specifically, it has a value of $1$, which roughly represents that it gives an advantage of 1 move to Lefty. (Games with more complicated shapes can have non-integer values, though!)

a Domineering rectangle, with width 1 and height 2

If Rita always has a strategy that will win, no matter who goes first, then the game has a negative value. For example, this Domineering game, where Rita can make a single move instead of Lefty, has a negative value. Specifically, it has a value of $-2$, which roughly represents a 2-move advantage for Rita. Notice how if she plays it well, she can make 2 moves in this space.

a Domineering rectangle of width 4 and height 1

If it matters who goes first, then the game is neither positive nor negative:

If the second player always has a strategy to win, then the game's value is $0$. For example, in this Col game, whoever goes first can color one of the cells, then the second player colors the other cell, and the first player is unable to move.

a Col game of two adjacent regions

If the first player always has a strategy to win, then we have a value which is not positive, negative, or zero, and so cannot be represented as a traditional number. An example is this Domineering game, where either player can move just once:

Domineering board, a 2 by 2 rectangle with the bottom right square removed

Games can be added together, which sums their values. Imagine you have two or more games next to each other. Two players alternate making moves, which they can choose to do in any of those games. The player who first cannot make a move in the combined game, i.e. cannot find a move to make in any of the individual games, loses. For example, if we add two copies of this L-shaped Domineering game to this 1-move (and value $1$) Hackenbush game, we get a game with the value $0$: it is a second-player win. (The first two moves, in whichever order, will be in the corners of the two Ls, leaving one move available for each player, and whoever started first will run out of moves first.) This shows that the value of that L-shaped Domineering game is $-\frac{1}{2}$.

Domineering board, an L-tetromino of width 3 and height 2, then Domineering board, an L-tetromino of width 3 and height 2, then Hackenbush of a single blue branch

Reversing a game, or taking its negative (and the negative of its value), is done by taking the game in which the moves available to the two players have been switched. For Domineering, reversing a game can be done by rotating it 90°. For Col and Hackenbush, switch the colors. For Frogs and Toads, take the mirror image, and ignore the art difference between frogs and toads.

In general, games can be compared, so that we can say which one (and its value) is greater or lower. Often, the individual values of two games will tell you numerically which is greater; for instance, a game whose value is $2$ is greater than a game with a value of $1$. In general, $G_1> G_2$ means that $G_1+ (-G_2)$ is positive (that is, Lefty wins). But not all pairs of games can be successfully compared. If $G_1+ (-G_2)$ is a game which the first player always has a strategy to win, then we can't say that $G_1$ is greater, and it's not lower either. You just have a pair of games that are not comparable. For example, let's look at a comparison between this 1-move Hackenbush game and this L-shaped Domineering game:

Hackenbush of a single red branch next to Domineering board, an L-tetromino of width 3 and height 2

To judge this, let's play the $G_{1}+ (-G_{2})$ game:

Hackenbush of a single red branch next to Domineering board, an L-tetromino of width 2 and height 3

Lefty can only move in the Domineering game, and his best move is in the corner (otherwise he leaves a move for Rita). If he goes first, Rita is left with a move in the Hackenbush game, which ends the game. If Rita goes first, she also starts in Domineering, leaving a move for Lefty there, and she still finishes the game with the Hackenbush move. Rita wins in both scenarios, and so $G_1+ (-G_2)$ is negative. $G_1+ (-G_2) < 0$ , which means that $G_1 < G_2$. And this fits with our independent knowledge of their numerical values: $G_1 = -1 < -\frac{1}{2} = G_2$.

Hackenbush of a single red branch next to a Domineering board, an L-tetromino of width 3 and height 2

Now that you have learned so much about Lefty and Rita's games, their classmate Maz is giving you a challenge. He has made a maze for you to follow. Using knowledge of the values of games, you need to navigate the grid below, starting with a green pen color, moving up along the green start arrow.

a hexagonal maze, with various games corresponding to each row of the grid

Maze rules:

  1. Each cell can be identified by a pair of coordinates, one (with blue arrows) on the left and upper left edges of the grid, and one (with red arrows) on the right and upper right edges of the grid. Each coordinate has a game position associated with it.

  2. If you enter a cell that has only one other exit, take that exit automatically and move towards the center of the next hex. Otherwise, you will try to compare the games at the two coordinates, to see which (if any) is bigger. (Except as amended by rule 5 below.)

  3. What direction do you turn? Which game is bigger? That is, if the game at the left coordinate is greater, turn left; if the game at the right coordinate is greater, turn right. If neither left nor right is the answer, then do not move: stay in place -- and change to a different pen color.

  4. Your turn should be as small as possible. That is, if you are supposed to be turning left or right, and there is an exit at a 60° turn in that direction, make the 60° turn. You will only make a 120° turn if the 60° turn is blocked by a black wall.

    a diagram: make a shallow turn if possible, otherwise, make a sharp turn
  5. If you need to determine a turning direction for a cell where you've already compared the two games, naturally the calculation based on the coordinates wouldn't change. To keep things fresh, if you repeat a cell, take the next game from the "games shelf" below, and add it to the (original) left coordinate for this comparison. (That addition is not permanent, but only for the purposes of this single comparison.)

No comparison will be between two games of exactly the same value.

"Games shelf": Each time you are doing a comparison in a cell you've done before, take the next game from this list, in order, and add it to the game from the (original) left coordinate. Each game will be taken down from the shelf only once.

Hackenbush stack, red blue blue from the base
Col game, three regions in a line, the left one is red
Hackenbush game, blue branch at the base, and three red branches each connected to it
Toads and Frogs game, toad toad blank frog frog
Domineering U pentomino, facing upwards
Col game, a three-circle Venn diagram with two two-circle intersections colored blue
Domineering hexomino, an I tetromino with a block to the left of the second square and a block to the right of the third
Hackenbush stack, blue blue red red from the base
Domineering game, a rectangle of width 2 and height 4
Toads and Frogs game, toad blank toad frog toad frog frog