Counting tiles in Fives and Threes: what they cannot hold

A double-six set has 28 tiles, and the set never changes. That is the whole basis of counting in Fives and Threes: you know exactly which tiles exist, you can see your own and those on the table, and everything else is somewhere in your opponent's hand or the boneyard. You do not need to memorise anything. You need one number, seven, and a habit of subtracting.

Why seven

List the tiles that carry a given number, say 4: 4-0, 4-1, 4-2, 4-3, 4-5, 4-6 and the double 4-4. That is seven tiles. The same holds for every number from 0 to 6, because each number pairs once with each of the six others and once with itself. Seven numbers times seven tiles is 49, which is more than 28 because a plain tile such as 4-6 is counted under both of its numbers, and 49 less the 21 plain tiles counted twice gives 28. Two facts follow.

• Every number is on 7 tiles.
• Every number shows on 8 tile ends, because the double has the number at both ends. The 28 tiles carry 56 ends in all, which is 7 numbers times 8.

The first figure is the one for counting tiles. The second matters when you count spots rather than tiles.

The subtraction

For any number, start at 7 and take away every tile with that number that you can see: those in your hand and those in the line. What is left is the count of unseen tiles with that number, and they are all either in your opponent's hand or in the boneyard.

Here is a worked position. The common two-player deal is seven tiles each, with the rest in the boneyard; Fives and Threes Dominoes lets you choose 6, 7, 8 or 9 tiles per player, and the counting works the same whatever the deal. You hold 6-6, 6-3, 5-5, 5-2, 3-0, 2-2 and 1-0. The line is 6-4, 4-1, 1-3, so the open ends are 6 and 3. Three tiles are on the table, which leaves 28 − 7 − 3 = 18 unseen tiles.

Across the seven numbers the unseen counts come out as 5, 4, 5, 4, 5, 5 and 4, for the numbers 0 to 6 in order. Check one: for 6, you hold 6-6 and 6-3 (two tiles) and the line has 6-4 (one), so 7 − 3 = 4 are unseen. Each row also adds back to seven: for 0, two in your hand, none in the line and five unseen is 2 + 0 + 5 = 7.

What a draw tells you

Now the opponent cannot match either end. In the app they draw from the boneyard until they can play, or pass if it is empty. A tile can only go on the 6 or the 3 if it carries one of those numbers, so at the moment they start to draw, they hold no tile with a 6 and no tile with a 3.

Count the unseen tiles that carry a 6 or a 3. They are 0-6, 1-6, 2-3, 2-6, 3-3, 3-4, 3-5 and 5-6: eight tiles. The other ten unseen tiles carry neither number, and that is where the opponent's hand must come from. Their hand cannot include any of these eight, so every one of them is in the boneyard right now. The drawing stops at the first tile with a 6 or a 3, and the boneyard is certain to contain at least those eight.

The chart shows what is left for the opponent number by number: how many of the ten “safe” tiles carry each number.

The bars add to 4 + 4 + 3 + 3 + 3 = 17 rather than 10 because most of the ten tiles carry two different numbers, and the doubles carry one. That is a reminder that the chart counts numbers, not tiles.

Using it

The point of the count is to choose what to leave exposed. With the opponent out of 3s and 6s, an end showing one of those numbers is closed to them for now, while the numbers 0 and 4 each sit on four of their possible tiles and are the easiest for them to answer.

• Leave the ends they cannot meet. If you play so that one end is still a 6 or a 3, they must draw again instead of replying. That is worth most when your own move scores, as in why 15 is the best total.
• Watch for a finished number. If you hold four tiles with a 5 and three more are in the line, all seven are accounted for and the opponent cannot hold a 5.
• Think about the 15. With two open ends, a 15 needs a double-six on a 3 or a double-five on a 5, as in doubles at an open end. If you can see the 6-6 or the 5-5 in your own hand or the line, the opponent cannot have it.

Counting tells you what is impossible, not what will be played. Before the opponent draws you know nothing about their hand beyond the pool. After a draw you know a fixed set of tiles it cannot contain. Use that, and check the scoring arithmetic in how Fives and Threes scoring works for the moves themselves.