How do you keep track of which dominoes have been played?

You do not memorise the tiles. You count numbers. In a double-six set every number appears on exactly seven tiles, so for any number you care about, you subtract the ones on the table and the ones in your hand from seven, and what is left is out there somewhere. Add a short list of who has passed on what, and by the last few turns of a four-player hand you can often name every tile that has not been played and say who holds it.

That sounds like a lot. In practice it is three small habits, done in order of importance, and the line of tiles on the table does half the work for you.

The line is your record

In line games the tiles that have been played stay face up on the table. You do not need to remember that a tile was played — you can see it. What you need is a fast way to read the line, and a memory for the two things the line does not show: who played each tile, and who passed.

Start with the arithmetic of the set. A double-six set has 28 tiles. Each number, blank to six, sits on seven of them: its double and one tile pairing it with each other number. Because the double carries the number twice, each number shows eight times as a half across the set.

That gives a quick check you can do by eye on a straight line. Every place where two tiles meet shows the same number on both sides of the join, so a join uses up two halves of that number. Every open end shows one. So for any number:

halves on the table = 2 × (joins showing it) + (ends showing it)

Take this line:

[2-4] [4-4] [4-6] [6-6] [6-3] [3-4]

The ends are a two and a four. Fours meet at two joins — either side of the [4-4] — and one end shows a four. That is 2 × 2 + 1 = 5 halves of the eight. Three are left. The [4-4] is already down, so those three halves are three separate tiles: [4-0], [4-1] and [4-5]. If you hold [4-5], exactly two fours are in other hands.

The double matters for that last step. While a number's double is still out, two of the missing halves belong to that one tile, so the tiles left are the halves left minus one.

The join count only works on a single line with no spinner. In All Fives and other games where a double opens up to four branches, count the tiles instead — it is just as quick once you are used to it.

What to count, in order

Nobody counts all seven numbers every hand, and you do not need to. Count in this order and stop where your attention runs out.

1. The numbers on the ends right now. These are the only numbers anyone can play. Before every turn, know how many of each end number are still out. When the answer is one, that end is nearly closed — Jamaican players call it a hard end — and whoever holds the last tile controls it. When the answer is zero, the end is dead.

2. Your long suit. If you hold four fives, you want fives on the ends, and you need to know when the rest of the table has run out of them. The guide to what makes a good player explains why holding a suit is worth more than spending it.

3. The doubles still out. There are seven doubles, and a double can only go down on its own number. Once the other six tiles of a number are gone, the double can only be played if that number is showing on an end, and one play can shut it out for the rest of the hand. Spanish-language partnership guides call this ahorcar — hanging a double. A guide based on Ignacio Zaibert's book on partnership dominoes advises trying to locate the big doubles early, by noting who could have played one and did not.

4. Your partner's tiles. The same guide's advice is to remember the tiles played, above all your partner's, and the moment each was played. A partner who avoided a number they could have followed has told you something.

Keep a pass list

A pass is the only information you cannot recover by looking at the table, so it is the thing to hold in your head. Keep it as a short list per player:

• Left opponent: no 5, no 3
• Partner: no 0
• Right opponent: nothing yet

In a block game a pass is certain: the player holds no tile matching either end, and since nobody draws, that stays true for the rest of the hand. In draw games it depends on the table. Where drawing while you can play is forbidden, as in the Puerto Rican rules recorded at pagat.com, a draw proves the player held nothing playable. Where drawing is allowed at any time, as in pagat's main draw rules, a draw proves nothing. The guide to reading a pass goes through both.

Putting it together late in the hand

Counting pays off at the end, when few tiles are left and each one matters. Take a four-player partnership hand with all 28 tiles dealt, seven each. You have counted, and five tiles are unaccounted for:

[5-0] [5-2] [3-0] [2-2] [1-0]

You know how many tiles each player has left because you watched them play: the opponent who plays after you has two, your partner has one, the other opponent has two. Your pass list is the one above.

Work through it:

• The opponent after you has passed on fives and threes. That rules out [5-0], [5-2] and [3-0]. Their two tiles must be [2-2] and [1-0].
• Your partner has passed on a blank. That rules out [5-0] and [3-0]. Your partner's one tile is [5-2].
• The other opponent holds what remains: [5-0] and [3-0].

Every tile in the game is now placed. Say the ends show a five and a one and you hold [1-3] and [5-6]. Play [1-3] on the one and the ends become five and three — the opponent who plays next is certain to knock. You did not guess; you counted.

Not every hand resolves this cleanly. Often you get to "one of these two tiles is with my partner", and that is still enough to decide most plays.

Count the pips before a block

The same counting tells you whether a block is good for you. A double-six set holds 168 pips. Subtract the pips on the table and in your own hand, and the rest is spread across the other three hands.

In the example above, the other hands hold 5 and 8 pips for the opponents and 7 for your partner. Your own two tiles are 15. Whether a block helps depends on how your table settles one. Where the team with fewer pips wins — the rule pagat records for Latin American partnership play — your side has 22 against 13 and would lose. In Jamaican partner play, as pagat describes it, the single player with the fewest pips wins, and here that is an opponent with 5. Either way, you would not want to close this hand. The guide to blocked games covers how each tradition decides.

The Zaibert-based guide makes the same point from the other side: if you pick up heavy tiles, start counting pips early, because a closing can come before you have shed them.

Boneyards and bigger sets change the numbers

Everything above assumes all 28 tiles are dealt. When some sit face down, the count still works, but "unseen" now includes the boneyard, so it tells you what could be in a hand rather than what must be.

In two-player block as pagat describes it, each player gets seven and fourteen tiles sleep. Any tile you cannot see is in your opponent's hand only one time in three. The guide to two-player dominoes works through what that does to strategy.

Bigger sets change the base number. In a double-nine set each number is on ten tiles and shows eleven halves; in double-twelve, thirteen tiles and fourteen halves. The guide to domino sets has the full arithmetic.

How to practise

Counting is a habit, not a talent. Two drills build it quickly.

Solo drill. Shuffle a double-six set, take seven tiles, then turn the rest up one at a time. After each tile, say how many of your longest suit are still face down. Check yourself every few tiles.

At the table. Count one number per hand — the one you hold most of — for a week. Then add the end numbers. Then add the pass list. Most players who try this find that within a few sessions they notice the moment they hold the last tile of a number, and start playing differently because of it.

The guide to how much of dominoes is luck shows why these small certainties add up over a match.