Two dice odds, and what a spot on the board is really worth
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Type the numbers on the tiles that touch each spot. Leave a field empty if the spot only touches two tiles, or if one of them produces nothing.
Results
| First spot | Second spot | |
|---|---|---|
| Ways out of thirty six | — | — |
| Yield per roll | — | — |
| Rolls that produce something, in per cent | — | — |
| Rolls that produce nothing, in per cent | — | — |
The two middle rows are not the same measurement, and keeping them apart is the whole point. Yield counts every tile that would pay out, so a number appearing twice counts twice. The row below it counts rolls instead, and a roll only happens once no matter how many of your tiles are waiting on it.
That is why a repeated number is not the upgrade it looks like. Two tiles on the same eight pay double whenever it comes up, and come up exactly as rarely as one of them would. You receive more, less often, and a plan built on that spot spends longer waiting.
The starting values are there to make a second point. Six and eight next to a two look far better than five, nine and ten, and they are worth precisely the same: eleven ways out of thirty six either way, silent equally often. Judging a spot by whether it holds the two best numbers ignores the third tile, and the third tile is where the difference usually is.
Every total on two dice
| Total | Ways to roll it | Chance, in per cent |
|---|
The table is symmetrical around the bold row, which is why the numbers on either side of it pair up. A six is worth five times a two, and stepping from a six down to a five costs a fifth of the production rather than a small nudge, because there is one fewer way to make it out of five.
What none of this measures is what the tiles actually produce. A spot with fewer ways can still be the better one if it gives you something you cannot get elsewhere, and a spot with more ways is worth less if all three tiles pay the same thing. The arithmetic settles how often, and leaves what to you.
Why are two spots that look different worth the same?
Because what a spot is worth is the number of ways its tiles can be rolled, and the eye compares the best single number instead of the sum. A six and an eight next to a two come to eleven ways out of thirty six, and so do a five, a nine and a ten.
The pair that catches your attention is only two of the three tiles. The third one decides more arguments than the first two, precisely because nobody looks at it.
| Tiles | Ways out of thirty six | Silent rolls, in per cent |
|---|---|---|
| 6 8 2 | 11 | 69,44 |
| 5 9 10 | 11 | 69,44 |
| 8 8 3 | 12 | 80,56 |
| 6 8 5 | 14 | 61,11 |
Is a repeated number good or bad?
It raises what you collect and does not raise how often you collect. Two tiles waiting on the same eight pay twice whenever it comes up, and it comes up exactly as rarely as it would for one of them.
So a spot with a repeat can beat one without on average while spending longer producing nothing at all. Whether that suits you depends on whether you need a steady trickle or a large occasional payment, which is a question about your plan rather than about the dice.
How much better is a six than a five?
There are five ways to roll a six and four to roll a five, so the six produces a quarter more often. Said the other way round, stepping down from a six to a five costs a fifth of that tile's production.
The same step near the edge of the table is far more brutal in relative terms: a three has twice the ways of a two, so those two neighbours differ by a factor rather than by a fifth.
Why is seven the most common total?
Because it is the only total that every face of the first die can reach. Whatever the first die shows, exactly one face of the second completes a seven, which gives six ways out of thirty six.
Every other total loses ways as it moves away from the middle, which is why the table is symmetrical around it and why a two and a twelve are the rarest things on the board.
Does the spot with more ways always win?
No, and the page does not claim it. It measures how often the tiles pay, not what they pay, and a spot with fewer ways can be worth more if it gives you something no other spot can.
It is also blind to everything that is not a roll: what your opponents took, where the board is blocked, and what you plan to build. What it settles is the how often, exactly, and that is usually the part being guessed at.
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