Every Fives and Threes score comes from adding spots, so it helps to know how many spots exist and where they sit. A double-six set has a fixed shape. It has 28 tiles, 168 spots in all, and you can rebuild every figure below from one rule: a tile is a pair of numbers from 0 to 6, and each pair appears once.
Where 28 comes from
A tile has two halves, and each half shows a number from 0 (blank) to 6. Split the tiles into two kinds.
- Doubles have the same number on both halves: 0-0, 1-1 and so on up to 6-6. That is 7 doubles.
- Plain tiles have two different numbers. Choose two different numbers from seven and you get 7 × 6 ÷ 2 = 21, because 7 × 6 counts each pair twice (4-6 and 6-4 are the same tile).
Together that is 7 + 21 = 28 tiles. Between them they have 56 halves, since each tile has two.
Where 168 comes from
Add up the spots on a tile and you get its total. The doubles have totals of 0, 2, 4, 6, 8, 10 and 12, which add to 42. The 21 plain tiles hold the rest. There is a shortcut for the whole set.
Each number appears on 8 halves: the 6 plain tiles that carry it show it once each, and its double shows it twice, so 6 + 2 = 8. A number n therefore contributes 8 × n spots, and the seven numbers contribute 8 × (0 + 1 + 2 + 3 + 4 + 5 + 6) = 8 × 21 = 168. Two checks agree. The plain tiles hold 168 − 42 = 126 spots, which is 6 per tile on average, and the 56 halves average 3 spots each, giving 56 × 3 = 168.
The same average gives a quick sense of a hand. The 28 tiles average 168 ÷ 28 = 6 spots, so a hand of seven tiles holds about 42 spots in the long run. Any one hand will differ, but a hand far above that is heavy in high numbers and one far below is light.
Seven tiles, eight ends
This is the count that counting tiles in Fives and Threes uses to work out what an opponent cannot hold. Every number is on 7 tiles, but it shows on 8 halves, because the double carries it twice. The two figures are easy to mix up, so keep them apart:
- When you count tiles that carry a number, the answer is 7 for every number.
- When you count spots or halves, the answer is 8 for every number.
The line matters here too. In Fives and Threes Dominoes a double at an open end counts both its spots, so a 4-4 on the end is worth 8 to the total, not 4. That is the 8-halves rule seen from the table: the double is the one tile that puts a number’s weight on a single end twice over.
How many tiles show each total
Now sort the 28 tiles by total. A total from 0 to 12 is made by pairs that add up to it, with the smaller number first so that each tile is counted once.
- Total 0: 0-0 (1 tile).
- Total 1: 0-1 (1).
- Total 2: 0-2, 1-1 (2).
- Total 3: 0-3, 1-2 (2).
- Total 4: 0-4, 1-3, 2-2 (3).
- Total 5: 0-5, 1-4, 2-3 (3).
- Total 6: 0-6, 1-5, 2-4, 3-3 (4).
- Total 7: 1-6, 2-5, 3-4 (3).
- Total 8: 2-6, 3-5, 4-4 (3).
- Total 9: 3-6, 4-5 (2).
- Total 10: 4-6, 5-5 (2).
- Total 11: 5-6 (1).
- Total 12: 6-6 (1).
The counts run 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1. They add to 28, and weighting each total by its count (0×1 + 1×1 + 2×2 + … + 12×1) gives 168, so the list and the spot count agree. The pattern is a mirror: swap every tile’s number n for 6 − n and a total of s becomes 12 − s, so the count for 5 equals the count for 7 and the whole row is symmetrical around 6.
Why the shape matters
Only the tile totals are counted above; in play the score depends on the two open ends, which are single halves, and a double at an end counts both spots. Even so, the set’s structure shapes what you see.
- Middle totals are plentiful. Ten tiles have a total of 5, 6 or 7 (3 + 4 + 3) against two tiles at 0 or 1, so a drawn tile is far more likely to be a middling one than a very light or very heavy one.
- Extremes are rare. Only 6-6 reaches 12, and only 0-0 is a blank pair. You can see in a moment whether either is gone.
- The boneyard is smaller than you might think. The app lets you choose 6, 7, 8 or 9 tiles each. With two players that takes 12, 14, 16 or 18 tiles out of 28, leaving 16, 14, 12 or 10 to draw from.
To turn totals into points you still need the multiples-of-three and multiples-of-five rule, set out in how Fives and Threes scoring works. The set is the stock you are scoring from, and knowing it exactly is what makes counting possible: 28 tiles, 7 doubles, 21 plain tiles, 168 spots, 7 tiles and 8 halves for every number.
Fives and Threes Dominoes lets you play solo 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.