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Two dice on a table beside a row of tally blocks Two square dice sit side by side on a table line. The left die shows three pips and the right die shows four, which add to seven. To the right of them a row of stacked blocks rises to a peak in the middle and falls away at both ends, which is the shape a tally of dice totals makes.

Seven Wins and Nobody Chose It

About 14 minutes

Find two dice and a piece of paper. Write the numbers 2 to 12 down the side, roll the pair, and put a mark next to whatever total comes up. Do it sixty times. It takes about four minutes.

The marks do not spread out evenly. The middle of the list fills up and the two ends stay nearly empty. Seven collects more marks than anything else, and 2 and 12 collect almost none.

That should bother you. Nothing on either die is bent. Every face has the same chance as every other face. So who decided that seven wins?

Nobody rolled a seven

Here is the move that unlocks the whole thing, and it is one sentence long.

A total is not a thing you roll. You roll two numbers. The total is something you work out afterwards, and several different rolls can give you the same one.

There is exactly one roll that gives you 12: a six and a six. There is exactly one that gives you 2: a one and a one. But a seven can arrive as 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, or 6 and 1.

That is six different rolls all landing on the same line of your paper. The seven is not luckier. It is just wider.

Two dice, or one die twice

Before counting, one thing has to be settled, because it is where nearly everyone goes wrong.

Is "1 and 6" the same as "6 and 1"?

For the total, obviously yes — both make seven. For counting the rolls, no, and it matters enormously. The easiest way to see it is to stop rolling two dice at once and roll one die twice instead. Now the two numbers are plainly separate: there is a first roll and a second roll, and "first a 1, then a 6" is not the same event as "first a 6, then a 1". They are two different things that happen to add to the same number.

Rolling two dice together is exactly the same, it is just harder to see. The dice do not know they are supposed to be a pair.

So: six faces on the first roll, and for each of those, six faces on the second. Six sixes. Thirty-six rolls in all, and every one of the thirty-six is as likely as every other one.

Now the counting is easy, because it is only counting.

All thirty-six rolls of two dice, arranged in a square A six by six grid. Along the top are the six faces of the first roll and down the left side are the six faces of the second roll. Each of the thirty-six squares holds the total those two rolls make. Squares holding the same total lie along a diagonal line, and the diagonals are longest through the middle of the grid where the total is seven, and shortest in the two opposite corners where the totals are two and twelve. Five of the totals are outlined so they can be chosen. 234567345678456789567891067891011789101112 123456123456 first roll second roll

Every square is one of the thirty-six rolls. Across the top is the first die, down the side is the second. Pick a total and see how many squares it owns.

  • A total of 2 — one square. You need a one, and then another one. There is no second way to do it.
  • A total of 5 — four squares: 1 and 4, 2 and 3, 3 and 2, 4 and 1. Four rolls out of thirty-six.
  • A total of 7 — six squares, the longest line on the grid. Every face on the first die can make a seven, because whatever it shows there is exactly one face on the second die that finishes the job.
  • A total of 10 — three squares: 4 and 6, 5 and 5, 6 and 4. Getting rarer as you head for the corner.
  • A total of 12 — one square, in the corner. Six and six, and nothing else.

The shape falls out on its own

Line the counts up and something appears that nobody designed.

Total 2 3 4 5 6 7 8 9 10 11 12
Rolls that make it 1 2 3 4 5 6 5 4 3 2 1

Up one at a time to six, then straight back down. It is a hill, and the hill has a reason: on the grid, each total owns one diagonal line of squares, and the diagonals get longer as they march towards the middle of the board and shorter as they leave it.

Turn those counts into percentages and you have a prediction you can check against your own piece of paper:

Seven is six times more likely than twelve. Not because it is favoured, but because it is six times easier to make.

What a board game already knew

Look at a Monopoly board. The first square you can land on after a corner is not the expensive one. The properties that get landed on most sit six, seven and eight squares past the jail, because that is where the hill puts you.

Backgammon is built on the same hill: the safe distance to leave a piece is the one that is hard to make with two dice, and every experienced player knows without being told that 7 is the worst gap to leave and 12 is nearly safe.

Neither game explains this to you. The counting was done long before the board was printed, and the board just quietly agrees with it.

You roll two dice. Which is more likely — a total of 4, or a total of 11?

The counting won, but only just

Go back to your sixty rolls and compare them with the prediction. They will agree roughly. They will not agree exactly, and yours will be lumpy in a way the smooth hill is not — some total will have done far better than it should, and another will have been badly cheated.

Sixty rolls is not many. Mine were worse than yours: in my first twelve rolls, seven — the most likely total there is — did not come up once, and twelve, the rarest, came up.

So the counting says one thing and a short run of real dice says something else. Both cannot be right about the same twelve rolls.

How many rolls does it take before the dice stop arguing?