Two different questions
In a roll-and-move game, “what are the odds of landing on that square?” can mean two things. One is the chance of reaching it with your very next roll. The other is the chance that your piece lands on it at some point as it travels past. The first is a plain dice question. The second needs a little more thought, because a piece can take many different routes, and each turn it may hop over the square you care about.
The next roll
Start with the simple case. If a square is exactly k spaces ahead, the chance of hitting it on the next roll is the chance that the roll equals k.
With one six-sided die, any square from 1 to 6 spaces ahead has a 1/6 chance, about 16.7 percent, and squares farther than 6 spaces cannot be reached in one roll.
With two dice added together, the picture changes. The chances follow the pattern of sums, where seven is the most common and two and twelve the rarest:
- 2 or 12 spaces ahead: 1/36 each
- 3 or 11 spaces: 2/36 each
- 4 or 10 spaces: 3/36 each
- 5 or 9 spaces: 4/36 each
- 6 or 8 spaces: 5/36 each
- 7 spaces: 6/36, or 1/6
A square one space ahead cannot be reached at all with a two-dice sum. This is why the way two-dice sums cluster around seven matters for board design: it determines which distances a piece is most likely to cover in a single move.
Eventually landing on a square
Now suppose the square is some distance ahead and you keep rolling until you pass it. Let u(n) be the chance of landing exactly on a square n spaces ahead at some point. For a single six-sided die, you can build the answer step by step. To land on square n, your last move must have started from one of the six squares just behind it, so u(n) is one sixth of the sum of the previous six values, with u(0) = 1 because you are already standing on your start square.
- u(1) = 1/6, about 0.167
- u(2) = (1/6) × (u(1) + u(0)) = (1/6) × (7/6) = 7/36, about 0.194
- u(3) = (1/6) × (u(2) + u(1) + u(0)) = (1/6) × (7/36 + 6/36 + 36/36) = 49/216, about 0.227
The chance rises as the square moves farther away, because more routes become possible. For the third square, a single roll of 3 is one route, but so are 1 then 2, 2 then 1, and 1 then 1 then 1.
The long-run answer
Keep going and the numbers level off. For a square far from the start, the chance of landing on it settles at about one divided by the average move. A fair six-sided die averages 3.5, so the long-run figure is 1/3.5, which is 2/7, or about 28.6 percent. Put another way, a piece using one die touches a typical distant square roughly 2 times in every 7 passes, and skips it the rest of the time.
With two dice the average move is 7, so the long-run figure is about 1/7, or 14.3 percent. On a loop of 40 squares, a piece would then be expected to land on any particular square about once every 7 laps, taking the game as a whole. These are long-run averages, not promises about any single trip around the board.
This pattern is a feature of the movement rule alone. It does not depend on which square you pick, as long as the square is well away from the start and the board has no special spaces.
When special squares change the picture
Real boards rarely stay uniform. Common rules that bend the numbers include:
- Go-to-start or send-back squares. A piece that is sent back will have more chances to land on squares near the start.
- Ladders and shortcuts. A square at the foot of a ladder is visited normally, but the squares the ladder skips are visited less.
- Doubles and extra turns. Rolling the same number on both dice may grant another roll, change the path or send a piece to a penalty square.
- Draw cards that move pieces. Instead of following the dice, they redirect a piece to a named square, raising its visit rate.
- Exact finishing rules. Needing an exact roll to finish makes the last squares sticky, so pieces linger there.
Each of these breaks the tidy “one divided by the average move” result. The way to handle them is to describe the board as a set of squares with probabilities of moving between them, which is the idea behind Markov chains. The question for a game like Monopoly is taken up in the article on its long-run square frequencies, where jail and card squares pull the numbers away from an even spread.
Estimating when the exact answer is hard
If the rules are complicated, a Monte Carlo simulation is often easier than algebra. Simulate a piece moving thousands of times, record each square it lands on, and divide by the number of moves. The results will be close to the true long-run frequencies, though not exact, and a larger run narrows the gap, in line with the law of large numbers. A spreadsheet or short program is enough for most family-game boards.
What this means for designers and players
For a designer, the useful point is that movement rules decide where the action happens. With one die, nearby squares are reached often and no square is out of range of a single move. With two dice, the middle distances dominate, and squares one space ahead are never reached directly. If an important space sits at a typical two-dice distance from another important space, players will meet it more often than a uniform board would suggest.
For a player, none of this predicts the next roll. Dice have no memory, and knowing a square is visited about one time in seven over many trips says nothing about what you will roll this turn. It does make it easy to see why some squares feel like they come up all the time, and others hardly ever.