Tuesday, February 16, 2016

Dots and Boxes variation

In grades 1, 2, 3, we played this variation of dots & boxes: Mathify the Squares Game.

I'm enthusiastic about this game, but can't resist a quick comment about the "mathification." Dots & boxes is already a mathematical activity, it doesn't need to be "mathified." This term implies confusing arithmetic and calculation with math, something I've written about elsewhere and, I hope, is clearly not implied by our blog.

In any case, I'll use the shorthand MD&B to refer to this dots and boxes variation.

Notes from playing in class

In class, we first introduced the kids to vanilla Dots & Boxes with a pre-printed grid of dots. We knew that it would be too much to play on a lattice covering the whole A4 sheet, but we thought a quarter of the grid would work. That turned out to be too big and the game started to seem monotonous to the kids as there was too much time spent on the opening (playing on squares that don't yet have any filled sides.)

We rectified this problem in 3rd grade and played on much smaller grids, with sizes between 7x7 lattices (which yield 6x6 squares) and 10x10.

After they were comfortable with the vanilla game, we introduced the product version with dice. In our case, we just had the players take turns and didn't give an extra turn when someone completed a square.

Some rule variations
There are some simple variations depending on how you deal with completed squares:

  • no extra turn (this was the version we played in class)
  • player adds another side to a different square with the same value. For example, say the dice are 2 and 4 (product 8) and the player fills a square. They also must add a side to another square with value 8.
  • player rolls the dice and adds a new side (a full extra turn)
  • player adds a side to any square (dice and square values ignored)
Of course, you could also make the extra turn optional instead of compulsory. You might also have some ideas about different ways to handle cases where there are no more free sides on squares with the required number value.

Probability questions

Dice games naturally lead to probability questions. Here were two that I really liked, based on scenarios from a recent game play:

What question are we asking?
What is the chance a player will get both the 20 and 36 boxes:

One great answer was 1/2. The reasoning: we are going to play until all boxes are filled and each of us have an equal chance to fill this box, so 1/2. This is not quite right, since the person who is about to roll has an advantage, but I thought it was an interesting interpretation of the question.

A 2x2 square
In this configuration, what is the probability that the next throw will allow the player to complete a box in the 4-30-25-15 zone?


A bigger D&B family

One of the reasons I thought this D&B variation was so cool is because of our games matrix. Whenever we play games, J1 and I talk about some key characteristics of the game, particularly the amount of randomness and the strategic complexity. These are not entirely independent dimensions, since a larger amount of randomness reduces the number or importance of each player decision, thus the strategic depth.

The mechanics of this game gave us some ideas about how to dial up or down the amount of randomness in this family of games. Here is our list of members of this family, roughly ordered from least random to most random:
  1. vanilla dots and boxes: no random element
  2. mash-up with product game: squares are still labeled, but players control the two factors using selectors (like in the product game) instead of using dice. This can be played with different collections of factors and different size boards (including board variations where a product appears multiple times or only a single time, where values are ordered or randomly distributed).
  3. half-way house: one factor is chosen by players moving a selector, the other is determined by a dice roll (either before or after the "free" factor is selected).
  4. MD&B game: as played in class and described in the first link
  5. MD&B game where each factor appears only once. This is a case of dialing up the randomness by reducing the strategic options of the players. 

Ideas for other games

I'm excited to see what other games we can modify use the underlying idea from the MD&B variation. To be clear: use numbers to label parts of the game and then constrain the players' actions based on a die roll to involve either the pieces with corresponding labels or board positions with those labels.

Three specific examples:
  1. Hackenbush variation where segments of the picture are numbered. This could nicely incorporate probabilities by putting values of the least likely dice rolls closer to the ground.
  2. Ultimate tic-tac-toe meets the product game: from Art of Math. This is an old post, but I just happened to see it when preparing this post.
  3. Dice chess. Here's the wikipedia article. For some reason, I often forget about this variation, even though it is a nice way to reduce the strategic complexity of vanilla chess for beginners and has some nice links with probability.
If you have some favorites, I would love to hear in the comments!

Playing through the MD&B version several times, we came up with these rule variations that are worth your consideration:

  1. Game stops as soon as someone rolls a value that can't be played (alternatives are to let that person roll again or have them pass their turn)
  2. Remove some of the randomness: (a) on your turn, you roll and play a side with the required value, but they opponent also plays a side with that value. As we played it, that means moves (without filled boxes) go: A, B, B, A, A, B, B, etc. (b) When a player fills a box, they can choose to re-roll both, either, or none of the dice for their extra turn.
I think our favorite was a combination of all three of these components. Mixing 2a and b, you have to be careful to keep track of whose turn it is, but it lead helped bring out elements of strategy and more thinking about probability.

*Update 2*
Game phases
Above, we talked about how A4 (or even 1/4 of an A4, which I guess is equivalent to A6) is too big for beginning players of Dots & Boxes. They found the game "boring." J1 and I talked about this experience and it led us to considerations around game phases: opening, middle game, and end game. These are terms we first learned in chess, and we found it useful to contrast the two games.
Here were some observations:

  • Opening: a lot of choices, not obvious how most of those choices link with "scoring" or the winning objective of the game. At this stage, there seems to be little interaction between the players (there is enough space that most of their actions either don't bring opposing pieces together or there is a lot of open territory).
  • Middle game: still many playing choices, increasingly direct conflict between players, interim objectives within the game become more clear and there are some chances for plays that either score or more clearly move closer to the overall game objective.
  • End game: significantly fewer choices for each turn than the other phases, either because there are fewer pieces (chess) or most territory has been claimed. At this stage, players are able to focus on the overall game objective, rather than interim objectives.
What we realized is that the larger playing area for D&B significantly increases the length of the opening. Because this phase is the least connected with capturing boxes, it is the hardest for beginning players to see how their choices ultimate lead to scoring and it is the phase with the most available choices on a turn.

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