Math in Board Games: Dice, Risk, and Fair Rules

Our copy of Snakes and Ladders has a torn corner and one missing counter, and it still produces arguments. A child climbs a ladder to within six squares of the finish, then lands on a snake and drops almost to the start. Somebody always says the game is unfair.

It is unfair, in a specific and measurable way, and the number is sitting there for anyone who wants it. The reference account of Snakes and Ladders notes that the game is classed as complete chance, that it can be represented exactly as a Markov chain, and that the 100-square version with 19 chutes and ladders takes an average of 35.8 spins to finish. Since no decision changes that average, the game is not a contest at all. It is a random number generator wearing a board.

The dice come first, so the dice decide

That is the useful habit this article is about. Before you argue about strategy, look at what the dice can produce, because the shape of that set constrains everything the rules can do. Start with one ordinary six-sided die. Six faces, six outcomes, each arriving one time in six. Flat. Boring. Completely even.

Now roll two and add them. There are still only six faces on each die, but there are thirty-six ordered pairs, and the sums are not evenly spread. Getting a 2 means rolling one specific pair, two ones, so it happens once in thirty-six. Getting a 7 means rolling any of six different pairs, so it happens six times in thirty-six. Same dice, wildly different odds.

Look at the shape. It rises, peaks at 7, and falls away in mirror image. Seven is six times more likely than two, and the whole list adds to 36 of 36, as it must. The reference article on dice puts the general point plainly: summing several dice produces approximations to a normal distribution, which is why designers reach for more than one.

This is the same multiplication of choices that our AND and OR rules guide formalises. Two dice is an AND: this face and that face. Thirty-six branches, then add up the ones that give the sum you want.

Where the distribution sits on the board

Now the clever part, and it is worth showing a child. In Catan, hexes are labelled with numbers and produce resources when that number is rolled. Because 6 and 8 come up five times in thirty-six, they are the valuable labels, and they are printed in red on the tiles. Because 2 and 12 come up once in thirty-six, they are nearly worthless.

Then look at 7, the single most likely sum. In Catan, the rules summary of Catan confirms what happens when it is rolled: the robber moves, one hex stops producing, and every player holding eight or more resource cards discards half. The most probable event in the entire game is a punishment. That is not an accident. A designer who understood the table put the brake exactly where the dice will push hardest, so the game cannot run away.

This is why the rules are worth reading as mathematics rather than as history. Every number token is a probability label, and moving one token changes who wins in the long run, before a single decision has been made.

Risk is a choice with a printed price

There is a second word in the title, and board games handle it better than most classrooms do. Roll a 6 to win, or take a guaranteed two points and keep playing. That is what risk looks like at a table: a small certain gain weighed against a larger uncertain one, with the odds printed in the rulebook instead of hidden.

Because the distribution is known, the choice can be described rather than guessed. Needing a 6 on one die is one chance in six, about 17 percent. Needing a 7 on two dice is six chances in thirty-six, which is the same 17 percent. That coincidence is worth pointing out to a child, because it explains something they already feel. A 7 seems easy because it is the most likely sum, and it is still a one-in-six shot, exactly as hard as rolling a single 6.

A player who has counted the grid can say “that is about one in six” instead of “I feel lucky tonight.” Nothing about the dice changed. What changed is that the risk became a number they could argue with, which is the beginning of every sensible decision about risk anyone ever makes.

Balance and fairness are not the same thing

Here is where most family arguments get the vocabulary wrong, and a serious attempt at the Catan problem is worth quoting. A published analysis of Catan board balance opens by separating two ideas that sound identical: balance and fairness are not necessarily the same thing, and balance is the easier of the two to determine.

To measure balance, the author generated 100 million random board layouts, scored each one for how evenly resources and numbers are spread, and compared real boards against that pool. A board used in the final round of a 2016 national qualifier scored in the top 0.2% of all 100 million, which is a lovely result: the people who play seriously end up with a fairer board than chance would give them, without necessarily calculating anything.

The distinction is the whole lesson. Balance is a property of the board, and you can score it. Fairness is a claim about the players, and it is much harder, because a perfectly balanced board can still hand an advantage to whoever moves first. Our expected value guide asks a different question again: not whether players start even, but what a bet is worth. Here we are only asking whether the starting position is level.

How to test a game at your own table

You need paper, not a computer. Draw a six-by-six grid, label the columns 1 to 6 and the rows 1 to 6, and fill every cell with the sum of its row and column. That is the entire probability space of two dice, and a child has just built the table above with their own hand.

Then count. How many cells show a 7? Six. How many show a 12? One. Ask which sum should be the most valuable spot on a board, then check whether the game you own agrees. Our probability trees guide draws the same thirty-six branches as a tree instead of a grid, and either picture answers the question in under a minute.

The second test is about turn order, and it needs no theory. Play the same game twice with the same two people, swapping who goes first. If the first player wins both times, and again the next evening, you have found an imbalance that no amount of dice counting will fix. That is the fairness question, and it is measured by playing, not by computing.

What a child actually takes away

The phrase “that is unfair” usually means “I did not like how that turned out.” Board games let you replace the feeling with a count, which is a genuinely useful habit to build early. A rare result is not a cheating result. A run of bad luck is not evidence of anything, and the gap between theoretical and experimental probability is exactly what a single frustrating evening is measuring. That article notes that the textbook odds of two dice belong to a separate piece of writing. This is that piece.

Worth keeping the neighbouring subjects apart, too. Chess has no dice at all, so its mathematics is counting and pruning rather than probability. And a game theory treatment starts after the dice have stopped, asking how players reason about each other. This one never gets that far. It stays with the board, the tokens, and the thirty-six little squares that decide what the game can become.