When you cannot play in Fives and Threes, the boneyard is where the next move comes from. Whether that is a nuisance or a disaster depends on how many tiles in the pool would fit the open ends, and you can work that out from what you can see. This post counts two positions and shows why matching ends are harder to draw against than different ones.
Rules vary, so check yours
Dominoes tables do not agree on what happens when you have no play. Some draw one tile and pass if it does not fit. Some draw until a playable tile turns up. Some do not draw at all and simply knock. Some tables leave the last two tiles in the boneyard. The Fives and Threes app lists its own rule: if you cannot play, you draw from the boneyard until you can, or pass when it is empty. The maths below is for drawing until you can play, and its first-draw figure also covers a one-draw rule.
What you can count
As in counting tiles, every number from 0 to 6 sits on exactly seven of the 28 tiles. Two different ends, say 4 and 2, are carried by 7 + 7 − 1 = 13 tiles, because the 4-2 is counted under both numbers. If both ends are the same number, only 7 tiles carry it.
Subtract what you can see. Tiles in the line are gone, and so are any in your hand. If you cannot play, none of your tiles fits the ends, so none of the playable tiles is yours. Every remaining playable tile is either in your opponent's hand or in the boneyard, and from where you sit the boneyard is simply a random sample of those unseen tiles.
That gives one formula. If U tiles are unseen and k of them would fit, the chance that your next draw fits is k ÷ U.
Position one: ends 4 and 2
Suppose each player was dealt 7 tiles, the opponent opened with the 4-2, and you hold nothing with a 4 or a 2. The line has 1 tile and you hold 7, so 28 − 1 − 7 = 20 tiles are unseen: 6 in the opponent's hand and 14 in the boneyard. Of the 13 tiles carrying a 4 or a 2, one is the 4-2 in the line, so k = 12.
The first draw fits with probability 12 ÷ 20 = 60%. If it misses, 8 non-fitting tiles remain among 19, and the second draw fits with probability 12 ÷ 19 (about 63%), so the second draw is the first success 40% × 63.2% = 25.3% of the time. Over two draws you are playable 60% + 25.3% = 85.3% of the time. Only 8 tiles do not fit, so a boneyard of 14 must hold at least 6 that do.
The expected number of draws until the first fit is (U + 1) ÷ (k + 1), a standard result for drawing without replacement. Here that is 21 ÷ 13 = 1.6 draws.
Position two: ends 4 and 4
Now the opponent opens with the double 4-4, so both ends are 4. The same hands and boneyard apply, and again you hold no 4. Seven tiles carry a 4 and one is in the line, so k = 6 out of the same U = 20.
The first draw fits with probability 6 ÷ 20 = 30%. The expected number of draws is 21 ÷ 7 = 3.0, about twice as many as before. The full pattern, found by multiplying the chance of each miss in turn, runs as follows:
About one deal in eight needs six or more draws. Under a one-draw rule you would pass 70% of the time.
There is a small chance that the whole boneyard misses. Taking 14 boneyard tiles from the 14 unfitting ones among 20 can happen in only one way out of 38,760 deals, or about 0.003%. It is real, which is why passing when the boneyard is empty is in the rules, but too rare to plan around.
A later position
The pool shrinks as the line grows, and the maths is the same. Suppose the line has 9 tiles, you hold 4, and the ends are 3 and 5. Then 28 − 9 − 4 = 15 tiles are unseen: 6 with the opponent and 9 in the boneyard. Thirteen tiles carry a 3 or a 5. If 7 of them are in the line (and none in your hand, since you cannot play), then k = 6 and U = 15. The first draw fits with probability 6 ÷ 15 = 40%, and the expected number of draws is 16 ÷ 7 = 2.3.
The lesson is the ratio. Once many tiles of an end number are played, k is small and draws get longer. An end whose number is nearly exhausted is also hard for your opponent to play, as in blocking to deny a score.
What this changes at the table
- Matching ends are harder to answer. Seven tiles carry the number against 13 for two different ends, and the first-draw chance falls from 60% to 30%. Leaving matching ends makes the line harder for both players to answer.
- Count before you play. If you can leave ends that are expensive for the opponent to answer, as with doubles at the ends in doubles at an open end, they will be the one reaching into the boneyard.
Fives and Threes Dominoes is a solo game against a computer opponent on Casual or Competitive, to a target of 61, 121, 181 or 241 points, with 6, 7, 8 or 9 tiles each. If you cannot play, you draw from the boneyard until you can, or pass when it is empty. There are no adverts or in-app purchases, and optional match points are in-game only with no cash value.