Advanced Block Dominoes Tactics: Controlling Both Ends

A line of Block Dominoes has exactly two open ends, and every legal move changes one of them. That is the whole of the decision space: which tile, and which end. The plays that separate a decent player from a strong one come down to what the two ends will show after your tile goes down, and who can use them. Each decision below is worked through with the tiles on the table.

Count before you leave a number on an end

Each number from 0 to 6 appears on seven tiles: six mixed tiles and its double. Every one of those seven is on the table, in your hand, or unseen. So the count for any number is seven, minus the tiles on the table that carry it, minus those in your hand that carry it. Only unseen tiles can change an end against you.

Where the unseen tiles are depends on the seat count. With four players all 28 tiles are dealt, so every unseen tile sits in an opponent’s hand. With two players, 14 tiles are set aside before play and stay out for the whole hand, so most unseen tiles are in the pile. The game seats four on screens 640 pixels wide or wider and two on narrower screens, so the same count means different things on a phone and a desktop. The odds that a particular opponent holds one of the unseen tiles of a number are worked out in the mathematics article; here the raw count is enough.

Do the count before you leave a number on an end, not after. A number with no unseen tiles is an end nobody but you can touch. A number with five unseen tiles is an invitation, and three opponents each get a turn to accept it before you play again.

Closing the line

The line is closed when both ends show the same number. From that moment only tiles carrying that number can be played, by anyone. If you hold most of the remaining tiles of that number, a closed line hands you the initiative: opponents pass or spend their last tile of it, and you decide what the ends show next. If you hold few of them, you have given the initiative away.

In position 1 the 6-6 opened, as it always does with four players: the whole set is dealt, so somebody holds the highest double and must lead it. The ends show 4 and 5, and your 4-5 fits either. Played on the left, its 4 touching the 4, it makes both ends show 5. Played on the right, its 5 touching the 5, both ends show 4. Before choosing, count.

Fives: the seven tiles carrying a 5 are 0-5, 1-5, 2-5, 3-5, 4-5, 5-5 and 5-6. One is on the table, the 1-5. Three are in your hand, the 2-5, 4-5 and 5-5. Seven minus one minus three leaves three unseen: 0-5, 3-5 and 5-6. Fours: the seven tiles are 0-4, 1-4, 2-4, 3-4, 4-4, 4-5 and 4-6. The 4-6 is on the table and the 4-5 is in your hand, so five are unseen: 0-4, 1-4, 2-4, 3-4 and 4-4. The three opponents hold 28 minus 5 on the table minus 6 in your hand, which is 17 tiles between them.

Closing on 5 leaves three unseen tiles that can answer it, and you keep 5-5 and 2-5 to continue if nobody does. Closing on 4 leaves five unseen tiles that can answer it and you hold no other 4, so the first opponent with a 4 takes over.

Now the cost. The 4-5 was your only tile that fitted both ends, and it is gone. Until an end changes, your 0-2, 0-0 and 3-3 are frozen, because a closed line admits nothing but 5s. If the opponents keep passing you play 5-5, then 2-5, and then hold three tiles that need a 0, 2 or 3 to appear. A closed line that stays closed becomes a blocked hand, and a blocked hand is scored on pips. The consolation is that the frozen tiles weigh 2 plus 0 plus 6, which is 8 pips. Closing with 6-4, 6-5 and 3-3 in reserve would be a different decision.

Forcing a pass

A pass is forced when neither open end shows a number the player holds. The surest way to force one is to close the line on a number whose unseen count is zero, because then nobody but you can move. That is not a blocked hand: a hand blocks only when every player, you included, has no legal tile.

Count the 2s in position 2. The seven are 0-2, 1-2, 2-2, 2-3, 2-4, 2-5 and 2-6. Four are on the table: the 2-1 at the left end, the 6-2, the 2-2 and the 2-4. Three are in your hand: 0-2, 2-3 and 2-5. Four plus three is seven. No 2 is unseen, so the three opponents, holding 28 minus 8 minus 5, which is 15 tiles between them, cannot have one.

Play the 2-5 on the right, its 5 against the 5, and both ends show 2. All three opponents must pass, and the turn comes back to you with the ends unchanged. The count was complete, so the passes were certain.

What you do with the extra turn matters as much as earning it. Play the 2-3 on the right and the ends show 2 and 3, leaving you 0-2, 1-4 and 0-0, which is 2 plus 5 plus 0, 7 pips. The unseen 3s are 0-3, 3-3, 3-4 and 3-5: seven, minus the 1-3 and 3-6 on the table, minus the 2-3 you just played. Playing 0-2 instead leaves five unseen 0s (0-1, 0-3, 0-4, 0-5, 0-6) and 2-3, 1-4 and 0-0 in hand, which is 5 plus 5 plus 0, 10 pips, so 2-3 is better.

Either way, the left end is yours for the rest of the hand. Every 2 not on the table is in your hand, so no opponent can play there again, and you can hold the 0-2 back until it is the tile you go out with. An end only you can change is the strongest thing you can own.

A tile that fits both ends

Positions 1 and 2 both turned on a tile that could go at either end, and the method was the same each time. Work out what the two ends will show after each option, count the unseen tiles for the number you would be exposing, and look at what you would still hold that matches it. Prefer the option that leaves a number with few unseen tiles and a follow-up in your own hand. If the options are close, prefer the one that leaves a number an opponent has already passed on; what a pass proves is the subject of the article on reading passes. The same test applies when a tile fits only one end: count the number going on, not just the number coming off.

The last-tile decision

With two tiles left you cannot go out this turn, so the question is which tile to play so that the other is as certain as possible to play next turn. Two things decide it: how many unseen tiles can change the end you are relying on, and how many pips you would be caught with if the hand blocks.

In position 3 there are 28 minus 10 on the table minus 2 in your hand, which is 16 unseen tiles: two in the opponent’s hand and 14 set aside. Both your tiles are playable, the 0-1 on the left and the 3-6 on the right, and each play leaves the other as your last.

Play the 3-6 on the right, its 3 against the 3-3, and the ends show 1 and 6, with the 0-1 left in hand. Now count the 1s: 0-1, 1-1, 1-2, 1-3, 1-4, 1-5 and 1-6. Five are on the table, the 1-5, 2-1, 1-6, 4-1 and 1-3. One is in your hand. Seven minus five minus one leaves a single unseen 1, the 1-1, and a double played on the 1 end still shows a 1. Nothing the opponent can do removes the 1 from the left end. Whatever they play, or if they pass, your 0-1 goes down next turn and you are out.

Play the 0-1 on the left instead and the ends show 0 and 3, with the 3-6 left in hand. The 3s are 0-3, 1-3, 2-3, 3-3, 3-4, 3-5 and 3-6. Two are on the table, the 1-3 and 3-3, and one is in your hand, so four are unseen: 0-3, 2-3, 3-4 and 3-5. Any of the four played on the 3 end takes your number away, and the opponent, not you, decides which end the 0-3 goes on. Four of the 16 unseen tiles beat you, the opponent holds two of those 16, and if the hand blocks you are holding 9 pips instead of 1. The 3-6 goes first.

The opponent holds two tiles and plays one per turn, so they cannot go out before your next turn; the only way to stop you is to change the end, and the count says whether they can. When the opponent is down to one tile, or neither of your last two can be made playable, the question becomes whether to block rather than race, which is worked through in the endgame article.

Testing it against the computer

The Hard opponent on this site plays by the same signals. On top of shedding doubles and heavy tiles, it adds 14 points to any move that leaves both ends showing the same number, a bonus that rises as fewer unseen tiles carry the number left on an end, and 15 points for each end showing a number an opponent has already passed on. Over 6,000 simulated hands it beats the Medium level, which counts nothing, 54.6 per cent of the time. To practise, set the difficulty to Hard below the table, keep a running count of one number at a time, and watch how often the line closes on you.