Why 15 is the best total in Fives and Threes

Most totals in Fives and Threes that score at all are worth between 1 and 4 points. Fifteen is worth 8, and the next best total, 18, is worth 6. The gap is not an accident of the rules. It follows from how the two scoring tests combine, and the same sum explains why you will see a 15 far less often than a 12 or a 9.

Why 15 pays twice

The rule is to add the two open ends, then score the total divided by three if it divides by three, plus the total divided by five if it divides by five. Both tests apply to 15: 15 ÷ 3 = 5 and 15 ÷ 5 = 3, so it scores 5 + 3 = 8 points. The full rule and every total are set out in how Fives and Threes scoring works.

Fifteen is the lowest number that three and five both divide into. The next such number is 30, and two open ends cannot reach it: with a standard double-six set a plain end shows at most 6 and a double at an end at most 12, so the largest total is 12 + 10 = 22. Fifteen is therefore the only total that can ever score both ways.

Points per spot

Divide the score by the total and a pattern appears. Any total that divides by three scores exactly one point for every three spots, which is a third of a point per spot. Any total that divides only by five scores one point for every five, a fifth of a point per spot. For 15 the two rates add:

1/3 + 1/5 = 8/15, or about 0.53 points per spot.

So 15 earns over half as much again as a 9 or an 18 for each spot on the table, and more than two and a half times as much as a 10 or a 20.

The same sum says that, for equal totals, threes are the better sort: a 12 scores 4, while a 10 scores 2.

How rarely it comes up

Count the ways to make each total. In a double-six set an end can show a plain number from 0 to 6, or a double that counts twice, 0 to 12 in steps of two. That is 14 different end faces. Pair them up, leave out the double-six against itself because there is only one double-six, and you get 98 distinct pairs of ends. Then add up the points over all 98.

Forty-eight of the 98 pairs score something, and they are worth 133 points between them, an average of about 1.36 per pair. The two pairs that make 15 are worth 16 of those points, so each is worth about six times the average. They are also only 2% of the pairs, which is why 15 feels like a prize when it arrives.

These are counts of pairs of faces, not chances: they ignore what has been played and who holds what.

The two pairs that make 15

Every 15 needs a double at one end, because two plain ends top out at 12. The double must be even, so the other end must be odd to make an odd total. That leaves exactly two pairs:

1. Double-six (12 spots) with a plain 3.
2. Double-five (10 spots) with a plain 5.

Spotting and denying a 15

Because a double has to be laid on a matching end, a 15 usually comes about in one of two ways.

• You lay the double. The ends show 6 and 3, and you hold the double-six: lay it on the 6 and the ends become 12 and 3. Likewise 5 and 5 with the double-five.
• You lay the plain end. A double-six is already sitting at one end. A tile that leaves a 3 at the other end scores 8; so does any tile that leaves a 5 against an exposed double-five.

The defensive reading is the same sum in reverse. Before you place a tile, look at the pair of ends it will leave. If it leaves 6 and 3, or 5 and 5, a double-six or double-five can still turn that into 8 points for whoever holds it. If you hold the double yourself, that is safe. If not, a placement that scores 2 or 3 but leaves such a pair may be worth less than a quieter one.

The reverse holds for an exposed double. An exposed double-six or double-five invites a 15 until it is covered, so count the 3s and 5s you hold before leaving one.

In Fives and Threes Dominoes the table shows the open-end totals and the points scored after each move, so you can watch for a 15 and see it score as 8.