Long suits: why many of one number controls the line

In a double-six set each number, from blank to six, is printed on exactly 7 of the 28 tiles. Players call the tiles carrying one number a suit, and four or five of them in one hand a long suit. This post counts how often you are dealt one, and what it does to the tiles the opponent can have, using a 7-tile hand for each player. The general method for working out what the opponent cannot hold is in counting tiles.

Seven tiles to a number

Take the 3s. The tiles carrying a 3 are 3-0, 3-1, 3-2, 3-4, 3-5, 3-6 and the double 3-3. That is six mixed tiles and one double, seven in all, and the same count holds for every number. The double is one tile, but it has two 3-ends. Each number therefore shows on 8 ends in total (six single ends plus two on the double). See inside a double-six set for the full count.

Two points to keep straight in everything below. A tile with two different numbers belongs to two suits at once, so the 3-5 counts as a 3 and as a 5. And the double counts once towards its suit, however many spots it shows.

How often you are dealt a long suit

There are two different questions here, and they have different answers. The first is about one named number, say the 5s. The second is about any number at all.

For a specific number, the chance of exactly k of its 7 tiles in a 7-tile hand is C(7,k) × C(21, 7−k) ÷ C(28,7), where C(28,7) = 1,184,040 hands. For four of them it is 35 × 1,330 = 46,550 hands, or 3.9%. Adding five, six and seven tiles, a specific number reaches four or more in 4.3% of hands, and three or more in 22.0%.

For any number, the chance is much higher, because there are seven numbers to hit. Counting every one of the 1,184,040 hands and taking the largest suit in each, a hand holds:

• 4 or more of some number in 349,356 hands, which is 29.5%.
• 5 or more of some number in 31,906 hands, which is 2.7%.
• 3 or more of some number in 93.0%, so a hand with no suit of three is the rare case.

A hand of seven tiles has 14 ends, which spread over seven numbers is 2 per number on average. So a suit of 3 is ordinary, a suit of 4 turns up in roughly three hands out of ten, and a suit of 5 is unusual.

What it does to the opponent

Suppose you hold k tiles of a number. The other 7 − k tiles of that number are among the 21 tiles you cannot see: the opponent's 7 and the 14 in the boneyard. The opponent's expected share is 7 × (7 − k) ÷ 21 tiles, which falls by a third of a tile for every one you hold. With k = 0 that is 2.33 tiles, with k = 4 it is 1.00, and with k = 6 it is 0.33.

What matters more is the chance of having none. If the open end shows that number and the opponent holds none of the suit, they cannot play there. They have to use the other end, or draw from the boneyard until they can, or pass when it is empty. The chance that the opponent's 7 tiles hold none of the remaining 7 − k is C(21 − (7−k), 7) ÷ C(21,7):

At 4 tiles it is better than one in four, and at 6 it is two in three: the one missing tile is in the opponent's hand 7 times in 21, so 14 ÷ 21. The figures assume you know only your own hand. Tiles on the line or already played change them by the same method.

What a long suit does not do

Holding many of one number does not let you keep an end on that number. When you play a tile on an end of 3, the new end is the other half of the tile. Play the 3-0 and the end becomes 0. The only tile that leaves the number unchanged is the double: the 3-3 on a 3 end leaves a 3 showing, and it counts both its spots towards the total, as covered in doubles at an open end.

So the benefit is about options, not ownership. With four 3s in hand, a 3 appearing at either end is an end you can play on, whichever way it turns up. The other three 3s are the only tiles anyone else can use there. That is a real advantage but a modest one: the opponent is shut out 27.4% of the time at four tiles, not every time.

There is also a cost. Tiles of one suit share a number, but their other halves are all different: 3-0, 3-1, 3-2, 3-4, 3-5 and 3-6 reach six separate numbers. A long suit therefore gives you many ways to leave a chosen total, which helps when aiming at 15 or away from a big reply, as in blocking. But four tiles of one number leave only three for everything else, and the hand can run short of plays when the ends move away from that number.

Using it at the table

• Count your suits at the deal. Tally the ends by number across your hand. Any number on 4 or more tiles is your long suit.
• Subtract from seven. Holding 4 of a number leaves 3. If 2 more are on the line, one is left.
• Keep your long suit's number in play when you can. That number at an end is a play for you and often not for the opponent. It saves a turn, but it is not a score in itself.
• Watch the draws. If the opponent draws, they had no tile for either end, so they hold none of either end's number.

All the figures here are for 7 tiles each. Other hand sizes change them, and the same formulas apply.