Glossary

The chance your next roll ends on each square, after any card you draw has finished moving you.

It is measured from wherever the piece is standing now, so it changes on every move.

A square with no colour cannot be reached this roll at all — not unlikely, impossible. Two dice move you two to twelve squares, so eleven squares are in direct range, plus wherever a card could send you from them.

This is the only colouring that looks forward. The other two say where turns end in general, which is why they share a colour scale and this one keeps its own.

Where turns end if you play forever — for each square, the share of all turns that finish there.

It does not change as you play. It is worked out from the rules rather than measured from a game, so it is the same answer whether or not anybody rolls.

Jail takes 11.5% of all turn endings, nearly four times any other square. Illinois Avenue leads the properties at 3.00%, then Go at 2.92% and B&O Railroad at 2.89%. The quietest three squares are the Chance squares, between 0.8% and 1.0%.

The same count as Expected, taken from the turns you have actually rolled instead of solved from the rules.

Both count where a turn ended, so both use one colour scale. That is deliberate: flip between them and a square running hot on your side is visibly redder than the same square on Expected. Give each its own range instead and ten rolls would paint their busiest square the same red as Jail, which tells you nothing.

One number for how far your run is from Expected.

It is the share of your turns sitting on the wrong square. At 40%, four turns in ten are in the wrong place. At 0% your run matches Expected exactly.

It falls as you roll, but not evenly: about 23% after 100 rolls, 7% after 1,000, and 2% after 10,000. A hundred times the rolls gets you ten times closer, so the last stretch takes far longer than the first.

Its formal name is total variation distance, if you want to look it up.

Two different counts. Landed on is every time you arrive, including arrivals you leave again immediately. Stayed on is turns that finished there.

They differ on eight squares. A Chance square gets about as many arrivals as its neighbours but keeps only three of every eight, because ten of its sixteen cards move you straight off it. Go To Jail is arrived at on 2.5% of turns and kept on none — no turn can end there. Jail runs the other way: you arrive on 5.8% of turns and are sitting there on 11.5%, because a sentence takes about two and a half turns to serve.

Everything the board colours is stayed on. Arrivals appear in the panel that opens when you point at a square.

The chance this roll puts you in Jail, counting all three routes at once: landing on Go To Jail, drawing a Go To Jail card from either deck, or rolling a third double in a row.

Where you stand decides most of it. From square 4 it is 0.35%. From square 23 it is 17%, because Go To Jail is exactly seven away and seven is the likeliest roll there is.

Carrying two doubles it never drops below 16.7%, whatever square you are on — a third double goes to Jail on its own, and six of the thirty-six throws are doubles. On square 23 with two banked it reaches 34%. That jump is what the middle of the board exists to show.

Roll a double and you get another turn. Roll a third in a row and you go straight to Jail without moving. The dots beside the piece count how many you have banked.

So on that third roll, six of the thirty-six throws are doubles and every one of them sends you to Jail. The other thirty move you normally — but not along the usual two-dice curve, because every double has been taken out of it. A total of 7 is untouched, since no double adds to 7. A total of 4 loses its 2+2 and becomes a third less likely.

In Jail you stay put and try for doubles for up to three turns. Fail all three and you pay the fine and move by that third roll. Rolling doubles gets you out but earns no extra turn.

The alternative is paying the fine straight away every time, and that is a different board — the choice moves Jail’s share by several points. The published frequency tables these figures match assume the same thing. A sentence works out at about two and a half turns.

Sixteen cards in each deck. Chance has ten that move you, including two separate “advance to the nearest railroad” cards plus a third naming the Reading Railroad — which is why the railroads run hot. Community Chest has only two movers: advance to Go, and go to Jail.

The six card squares sit at 2, 7, 17, 22, 33 and 36. The longest stretch of board without one is ten squares, and a roll covers eleven consecutive squares — so wherever you stand, at least one is in range.

Cards are drawn fresh each time, rather than from a pile that gets used up before it is reshuffled. Each card comes up equally often either way, so nothing measured here moves — but tracking a pile would make the honest answer for the next card 0% or 100%, depending on what you had already seen.

One card can land you on a square that makes you draw again. Go back three spaces from the Chance at square 36 puts you on the Community Chest at 33, so you draw from that deck too.

That is how “go back three spaces” ends up among the cards that can send you to Go: 1 in 16 to draw it, then 1 in 16 again for the Chest’s advance-to-Go, so 1 in 256. Routes like that name their middle square, because the first card on its own reads as nonsense.

Nothing chains further than that.

Square 10 holds two different situations. Land on it the ordinary way and you are just visiting, and your next roll is normal. Get sent there and you are in Jail, and your next roll is an escape attempt.

Of the 11.5% of turns that end on square 10, 9.4 points are locked up and 2.1 are passing through. The board colours the square by the total, because that is where the piece is standing either way.

Expected is worked out from the rules rather than estimated by playing millions of turns. It is the answer a simulation gets closer and closer to, taken directly.

What gets solved is every situation the piece can be in and the chance of going from each one to each other one. There are 123 situations: 40 squares times how many doubles you are carrying (none, one or two), plus three more for being in Jail on your first, second or third escape attempt.

Both halves are needed. Drop the doubles count and the three-doubles rule disappears; drop the jail count and you are modelling a piece that gets out on a fixed schedule. Either one shifts Jail’s share, which is the number the board is best known for.

It settles after about 220 passes over those 123 situations, and agrees with published figures to within 0.03 percentage points.

One roll. Rolling a double ends the turn and immediately grants another, rather than both rolls counting as one turn.

That is what published square-frequency tables count, which is what makes these figures comparable to them. It also means “rolls” and “turns” are the same number everywhere on this page.

No money, rent, property, houses or trading. None of it moves the piece, and where the piece goes is the only thing measured here.

One player, not several. Players never get in each other’s way in Monopoly — two pieces can sit on the same square, and nobody can be blocked — so one piece gives the same distribution as four.