Doubles at an open end in Fives and Threes

In Fives and Threes a double at an open end counts both its halves. A double-four sitting at the end of the line adds 8 to the total, not 4. That one rule makes doubles the most useful tiles in the set, and the reason is plain arithmetic rather than anything mysterious about the tiles themselves.

What the rule changes

Scoring itself is unchanged: add the two open ends, then score a third of the total if it divides by three and a fifth if it divides by five, as set out in how Fives and Threes scoring works. The only difference with a double is how many spots it contributes.

A double can only be laid on an end that shows its own number. Put the double-four on a 4 and that end goes from showing 4 to showing 8. So laying a double of number n on a matching end adds exactly n spots to the total, on top of whatever the line already had. A plain tile never does this: it swaps one number for another, and the total can go up or down.

Five placements, before and after

Each of these starts with one end showing the number of the double you hold. The other end stays as it is.

• Ends 6 and 3, total 9, which scores 3. Lay the double-six on the 6: the ends show 12 and 3, total 15, which scores 8.
• Ends 5 and 5, total 10, which scores 2. Lay the double-five on either 5: 10 and 5, total 15, which scores 8.
• Ends 4 and 1, total 5, which scores 1. Lay the double-four on the 4: 8 and 1, total 9, which scores 3.
• Ends 2 and 5, total 7, which scores nothing. Lay the double-two on the 2: 4 and 5, total 9, which scores 3.
• Ends 3 and 0, total 3, which scores 1. Lay the double-three on the 3: 6 and 0, total 6, which scores 2.

The double does not always help. Ends 4 and 3 total 7 and score nothing; the double-four makes 8 and 3, total 11, still nothing. A double adds a fixed amount, so it only pays when that amount lands the total on a scoring number. Run the sum before you lay it.

A double against every plain partner

To see how much the rule is worth, fix one end as a double and try it against each plain number from 0 to 6 at the other end. That gives seven totals, 2n + p, and you can add up the points. Do the same with a plain n in place of the double, total n + p, and compare.

The figures come from running all seven totals. For the double-six they are 12, 13, 14, 15, 16, 17 and 18; only 12 (4), 15 (8) and 18 (6) score, and 4 + 8 + 6 = 18. A plain six against the same partners gives 6 through 12, and 6 (2), 9 (3), 10 (2) and 12 (4) score, 11 in all. The double-six has the higher sum by 7.

It is not a straight climb. The double-four scores 9 and the double-three 11, because the extra spots land on a scoring total more often for the three. The doubles at 0 and 1 are worth the same as their plain tiles, 4 each, since an extra 0 or 1 spot rarely moves the total onto a scoring number. These are counts over equal partner ends, not chances: they ignore what has been played and who holds what.

The best partner for each double

Each double has one plain end it works best against. These are the totals to look for.

1. Double-six with a 3: 12 + 3 = 15, which scores 8.
2. Double-five with a 5: 10 + 5 = 15, which scores 8.
3. Double-three with a 6: 6 + 6 = 12, which scores 4.
4. Double-four with a 4: 8 + 4 = 12, which scores 4.
5. Double-two with a 5: 4 + 5 = 9, which scores 3.
6. Double-one with a 4: 2 + 4 = 6, which scores 2.
7. Double-blank with a 6: 0 + 6 = 6, which scores 2.

The first two are the only pairs that make 15, covered in why 15 is the best total. A double's contribution is always even, so the total has the same odd or even character as the plain end beside it. That is why the odd partners 3 and 5 pair with the double-six and double-five for 15.

Covering a double

A double counts twice only while it is an open end. Once a tile is played against it, the new end shows a plain number and the double sits inside the line, where it counts for nothing. Take the 15 above: double-six and a 3. Your opponent covers the double with the 6-2. The ends are now 2 and 3, total 5, which scores 1. The 8 points are not taken back, but the board that would have offered the next 8 is gone.

So an exposed double-six or double-five is an invitation to whoever can use it. Before you leave one open, count the 3s or 5s you hold, and remember a covering tile can change the total by far more than the tile you laid.

The same rule explains the opening tile: a lone double shows its full spots, so the double-six opens at 12 and scores 4, as in choosing your opening tile. In Fives and Threes Dominoes the table shows the open-end totals and points scored after each move, so you can compare the total you expected with the one on screen.