How Many Dominoes Are in a Set? Double-6, Double-9 and More

A standard double-six domino set (faces 0 to 6) has 28 dominoes, and a double-nine set (faces 0 to 9) has 55. For a set with faces 0 to n, the count is (n + 1) times (n + 2), divided by 2. The pattern is simple enough to check a bag of loose dominoes for missing tiles without laying them all out.

NRICH, the mathematics project at the University of Cambridge, asks this as its Domino Sets task: a bag of loose 0-6 dominoes needs checking for a full set, so how would you go about it, how could you be sure, and how many would a full 0-9 set contain? The counts and puzzles below were verified by listing each set in full.

1. Why the count is 28

List the dominoes by their larger face, so that no tile is counted twice. A tile with larger face 6 can pair 6 with 0, 1, 2, 3, 4, 5 or 6, which is 7 tiles. Larger face 5 gives 6 tiles, then 5, 4, 3, 2 and 1. The total is 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28.

The same sum for faces 0 to n is 1 + 2 + ... + (n + 1), which is (n + 1)(n + 2)/2. The table lists a few common sets (the arithmetic holds for any n).

2. Why the pips total n(n + 1)(n + 2)/2

Each face value appears n + 2 times across the whole set: n times on a mixed tile and twice on its own double. So the total of all pips is (n + 2) times the sum of 0 to n, which is (n + 2) times n(n + 1)/2. For the double-six set, that is 8 times 21, which is 168.

3. Puzzle 1: dominoes carrying a given number

How many dominoes in a double-nine set have a 4 on at least one end?

Answer: 10. The 4 can pair with each of the faces 0 to 9, which is 10 tiles, and the 4-4 double is one of them. The same count gives 7 dominoes carrying a 6 in a double-six set.

4. Puzzle 2: dominoes with no blank

A double-nine set has 55 tiles. How many have no blank (0) end?

Answer: 45. The tiles with a blank number 10 (0-0 through 0-9), so 55 minus 10 gives 45. That also equals the size of a set with faces 1 to 9, which is 9 times 10 divided by 2.

5. Puzzle 3: the missing domino

A bag should hold a full double-six set, but one domino is missing. The pips on the remaining 27 add up to 160. The missing tile is not a double and has no 6 on it. Which tile is missing?

Answer: the 3-5 tile. The full set has 168 pips, so the missing tile has 8 pips. The tiles that total 8 are 2-6, 3-5 and 4-4. The 2-6 has a 6, the 4-4 is a double, and the 3-5 satisfies both conditions. Listing all 28 tiles confirms exactly one candidate.

6. Puzzle 4: can the whole set form a closed ring?

Dominoes can be laid end to end so that touching halves match. Can all the tiles of a double-six set be laid in one closed loop?

Answer: yes, and it works whenever n is even. In a set with faces 0 to n, each face appears on n + 2 tile ends. In a closed ring, each face value must appear an even number of times (each match joins two halves). For n = 6, each face appears 8 times, which is even, so a ring is possible. For n = 5, each face appears 7 times, which is odd, so a ring is impossible for double-five. A search of every arrangement for sets from double-one to double-four agrees: double-two and double-four rings exist, while double-one and double-three rings do not. The parity argument is what rules out the odd case for good.

7. A different kind of domino puzzle

NRICH's Domino Square task takes the "double-3 down" set of ten dominoes (faces 0 to 3) and asks you to use them all to make a square in which each side has eight dots. The pip total helps here: the ten tiles carry 30 dots, but four sides of 8 make 32, so the four corner halves, which count on two sides, must add to 2. NRICH's student solutions also note that fixing one side at 8 and then flipping dominoes is what lets the other sides work out.

8. Key takeaways

• A set with faces 0 to n has (n + 1)(n + 2)/2 dominoes: 28 for double-six, 55 for double-nine.
• Count by larger face to avoid double counting: 7 + 6 + 5 + 4 + 3 + 2 + 1.
• The total pips are n(n + 1)(n + 2)/2: 168 for double-six, 495 for double-nine.
• A missing tile can be found from the pip total and a few conditions.
• A full set can form a closed ring only when n is even.