Starting with 6, 7, 8 or 9 tiles: what hand size changes

Fives and Threes Dominoes lets you start with 6, 7, 8 or 9 tiles per player. The set is a fixed 28 tiles, so the choice is a trade: tiles in your hand are tiles not in the boneyard. This post puts numbers on that trade for a two-player game, using only arithmetic you can repeat yourself.

What is left over

Two players are dealt n tiles each, and what remains is the boneyard: 28 − 2n. The table below also shows how much of the set is already in hands, and how many tiles are unseen from your side. Unseen means everything except your own hand, so it is the opponent's tiles plus the boneyard: 28 − n.

• 6 tiles each: 12 in hands, boneyard of 16, 22 unseen (the opponent's 6 and the 16 in the boneyard).
• 7 tiles each: 14 in hands, boneyard of 14, 21 unseen.
• 8 tiles each: 16 in hands, boneyard of 12, 20 unseen.
• 9 tiles each: 18 in hands, boneyard of 10, 19 unseen.

At 7 each, exactly half the set is dealt out. Each step up in hand size takes two more tiles from the boneyard, one from each player's side of the table. At 9 each, 18 of the 28 tiles, about 64%, are in hands before anyone has played.

The boneyard stops being a safety net

When you cannot play, the app has you draw from the boneyard until you can, or pass when it is empty. A big boneyard is a deep well to draw from. A small one runs out sooner, and a pass then costs you the turn.

The share of unseen tiles that sit in the boneyard is (28 − 2n) ÷ (28 − n):

• 6 each: 16 ÷ 22 = 72.7%
• 7 each: 14 ÷ 21 = 66.7%
• 8 each: 12 ÷ 20 = 60.0%
• 9 each: 10 ÷ 19 = 52.6%

From your seat, any unseen tile is equally likely to be in either place. With 6 each, a tile you cannot see is in the boneyard about 73% of the time. With 9 each it is barely better than a coin toss. That matters for the counting in counting tiles: a larger hand makes the opponent's hand a bigger part of what you cannot see, so what you learn from their draws and passes covers more of the tiles in play.

Being stuck becomes rarer

Every number from 0 to 6 sits on exactly seven tiles, as the double-six set post shows. A hand has no tile for a given number only if all n of its tiles come from the other 21. The chance of that is the number of ways to choose n tiles from 21, divided by the number of ways to choose n from 28:

• 6 tiles: 54,264 ÷ 376,740 = 14.4%
• 7 tiles: 116,280 ÷ 1,184,040 = 9.8%
• 8 tiles: 203,490 ÷ 3,108,105 = 6.5%
• 9 tiles: 293,930 ÷ 6,906,900 = 4.3%

Read this as the chance you cannot meet an open end showing that number from your starting hand, before any tile is played. Against two different ends, say 4 and 2, thirteen tiles fit, so only 15 tiles miss both. The chance of a hand with none of the 13 is 5,005 ÷ 376,740 = 1.3% at 6 tiles and 5,005 ÷ 6,906,900 = 0.07% at 9. Having a play at the start is nearly certain at every size. The difference shows up late, when the line has taken the tiles that fit.

The same 7-in-28 sum covers any group of seven tiles. The seven doubles are such a group, so the chance of holding no double at all is also 14.4%, 9.8%, 6.5% and 4.3%. Put the other way, 85.6%, 90.2%, 93.5% and 95.7% of hands hold at least one. Doubles are worth keeping in mind because of how they score at an open end.

More spots to carry

The set carries 168 spots over 28 tiles, an average of 6 per tile. A hand of n tiles therefore averages 6n spots: 36, 42, 48 and 54. Those are averages, not promises, since an actual hand can run well above or below. A double at an open end counts both its spots, so the 6-6 alone is worth 12 towards an end total, and a bigger hand has more chances of holding heavy tiles like it.

Longer hands, longer games

To go out you must play every tile, so the fewest turns possible is n: 6, 7, 8 or 9. That is a floor, not a typical length, because a draw adds a tile you then also have to play. The odds of finding a fit while drawing shift as the line grows, as drawing from the boneyard shows.

Choosing a size

• 6 tiles: the biggest boneyard (16) and the most chance of a hand with a missing number (14.4%).
• 7 tiles: the balance point, with half the set dealt and a 14-tile boneyard.
• 8 or 9 tiles: more choice each turn, a boneyard of 12 or 10, and a hand that rarely lacks a number (6.5% and 4.3%). You carry more spots, and at 9 the opponent's 9 tiles are 9 of the 19 you cannot see.

Whatever you choose, the sums above are the same ones to rerun as the game goes on: tiles seen, tiles unseen, and how many of the unseen would fit the ends.