Every scoring end total in Fives and Threes, 3 to 20

After every placement in Fives and Threes you add the two open ends and check the total. Between 3 and 20 there are 18 possible totals, and only nine of them score. This post lists those nine with their points, shows which pairs of ends make each one, and accounts for the other nine. The rule itself is covered in how Fives and Threes scoring works.

The rule, briefly

A total that divides by three scores the total divided by three. A total that divides by five scores the total divided by five. If both apply, add the two scores. A double at an open end counts both its spots. In the tables below, a plain number is an end showing that many spots, and D4 is a double-four at an end, which shows 8.

The nine scoring totals

With a standard double-six set a plain end shows 0 to 6 and a double at an end shows 0 to 12 in steps of two. The last column counts the pairs of end faces that give each total; a double-six cannot sit at both ends, because there is only one.

The points add up to 31 across the nine totals, and the pair counts add up to 48 of the 98 possible pairs. Why 15 stands apart, and what the counts mean for how often it appears, is worked through in why 15 is the best total in Fives and Threes, so this post does not repeat it.

The nine that never score

These are the totals between 3 and 20 that are not on the list:

• 4, 7, 8, 11, 13, 14, 16 and 17 divide by neither three nor five, so they score nothing, even though every one can be made with the right ends.
• 19 cannot be made at all. An odd total needs one plain end, which shows at most 6, so the other end would have to show 13 or more, and no end shows more than 12.

The totals 16 and 17 sit between two of the best scores on the table and pay nothing.

Which totals need a double

Two plain ends show at most 6 + 6 = 12. So any total above 12 needs at least one double at an end, and that covers three of the scoring totals: 15, 18 and 20. The other six (3, 5, 6, 9, 10 and 12) can each be made from two plain ends, which is why they come up in ordinary play.

The ceiling is easy to verify. The largest face is the double-six at 12 spots, the next is the double-five at 10, and the two cannot be beaten, so the highest possible total is 12 + 10 = 22. It divides by neither three nor five and scores nothing. The highest scoring total is therefore 20, and it needs two doubles at once: the double-six against the double-four, 12 + 8. That is the only pair that makes it.

The 18 has three routes. A plain 6 against the double-six is the simplest, and the other two both involve two doubles: double-three against double-six, and double-four against double-five.

Using the list

Three small checks cover most of it.

1. Multiples of three score a third of the total. Multiples of five score a fifth. The only total reached by both is 15.
2. For the same size, a multiple of three beats a multiple of five. A 12 scores 4 against 2 for a 10, and an 18 scores 6 against 4 for a 20.
3. Above 12, a total needs a double. If neither end is a double and neither is about to become one, the highest total you can make is 12.

When you choose between tiles, run the sum for each and compare. The nine totals above are the only answers that score, so any tile that leaves a total outside the list is worth nothing on that placement, however high the total looks.