Block Dominoes Endgame: Counting Down the Last Tiles

Why the last three tiles play differently

Nothing enters a hand of Block Dominoes after the deal. Nobody draws; a player who cannot match either open end passes, and the tiles set aside in a two- or three-player hand stay out for good (the rules page has the full ruleset). So each of the 28 tiles is always in one of three places: on the table, in your hand, or unseen. That makes the endgame a counting problem rather than a guessing one. With four players, every unseen tile is in an opponent’s hand. With two, fourteen of them were set aside and only the rest are held.

The other change is tempo. You play one tile per turn, so from three tiles you need three turns of your own, and between each of them every opponent moves. An opponent with one tile needs one suitable end, once. Every question below is a version of the same one: which of your plays makes your next turn certain, and if none does, which leaves you the lightest hand when the line locks.

Listing what is still out

The count is subtraction: 28, minus the tiles on the table, minus the tiles in your hand, equals the tiles you cannot see. Do it number by number. Each number from 0 to 6 sits on exactly seven tiles, six mixed and one double. Count the tiles on the table showing a 6 (the double is one tile and counts once), add the 6s in your hand, and subtract from seven: that is how many 6s are unseen. If the answer is zero, the number is dead, and an end showing a 6 can never be played on again by anyone. If the answer is one, a single unseen tile controls that end.

Dead and nearly dead numbers are the endgame’s main tool. The Hard computer opponent uses the same idea: part of its score for a move is a bonus that rises as fewer unseen tiles carry the number it is about to leave on an end. Both positions below give the full count.

Position 1: two players, two tiles each

This is the two-seat layout the game uses on narrow screens. Fourteen tiles were dealt, seven each, and fourteen set aside. Your opponent held the 6-6 and opened with it. Ten tiles have been played, in this order: 6-6, 6-3, 0-6, 3-3, 4-0, 3-5, 5-2, 2-2, then the opponent passed with the ends showing 4 and 2, then 2-1 from you and 1-1 from them. It is your turn.

First the count. Ten tiles are on the table and two are in your hand, so 28 − 10 − 2 = 16 tiles are unseen: 0-0, 0-1, 0-2, 0-3, 0-5, 1-4, 1-5, 1-6, 2-3, 2-4, 2-6, 3-4, 4-4, 4-5, 5-5 and 5-6. Fourteen of those are set aside and two are in the opponent’s hand, and from the count alone you cannot tell which. The pass sorts them. When the opponent passed, the ends showed 4 and 2, so they held nothing with a 4 or a 2 then, and since nobody draws they still do not (see reading passes). Eight of the sixteen carry a 4 or a 2: 0-2, 1-4, 2-3, 2-4, 2-6, 3-4, 4-4 and 4-5. All eight are set aside. The opponent’s two tiles are among the other eight: 0-0, 0-1, 0-3, 0-5, 1-5, 1-6, 5-5 and 5-6.

Now the decision. You cannot go out this turn, so you want the play after which your last tile is certain to fit next turn. Play the 1-3 on the right end. The ends become 4 and 3, and you hold the 4-6. The opponent holds no 4, so the 4 end cannot change. Whatever they do on the 3 end (only the 0-3 among their possible tiles fits it), the 4 is still showing when the turn comes back to you, and the 4-6 goes out. You win the hand and score whatever is left in their hand.

Play the 4-6 first instead and the ends are 6 and 1, with the 1-3 in your hand. Three of the eight tiles the opponent could hold fit the 1 end (0-1, 1-5, 1-6) and two fit the 6 end (1-6, 5-6). If they play the 1-6 on the 1, both ends show 6, the 1-3 is stuck and you pass; if their last tile is the 5-6 they go out, and if not, the hand blocks with your 4 pips against whatever they kept, and the 0-0, 0-1 and 0-3 all beat 4. Notice that the certain route keeps the heavier tile, the 10-pip 4-6, in hand. Low pips matter when a block is coming, and not at all when you are going out.

Going out, or blocking

Can you make your next turn certain? A play is certain when your remaining tile matches an end that no opponent can change. Position 1 used a pass to prove that; a dead number does the same job without one.

If not, who wins if the line locks? A hand is blocked when every player passes in turn. The lowest pip total wins it and scores the pips left in every other hand; a tie is settled by whatever house rule you agreed before play. You know your own total and you know the unseen tiles, so you know the floor of every opponent’s total: one tile weighs at least the lightest unseen tile, two at least the two lightest together. If your total is below those floors you cannot lose a blocked hand, and the only question left is whether you can force one.

If a block is coming and you may not win it, shed weight. Every pip you hold when the line locks counts against you, and a double is the heaviest tile of its number: the 6-6 alone is 12 pips and fits only one number. With two playable tiles and nothing certain either way, play the heavier one, and prefer a play whose new end is a nearly dead number, so that an opponent with a single tile is unlikely to find a home for it. Setting both ends to the same dead number locks the line on the spot. The tactics article covers closing the line earlier in a hand; in the endgame it can be the whole point, as the next position shows.

Position 2: four players, deciding to lock the line

Four seats, so all 28 tiles were dealt and nothing is set aside. Play goes from you to Left, then Top, then Right. Left held the 6-6 and opened with it. Twenty tiles are on the table; Left has one tile, Top has two, Right has three, and you have two. Top has passed once and Right twice. It is your turn.

The count: 28 − 20 − 2 = 6 unseen tiles, all of them in the three hands around you: 0-4, 1-3, 1-4, 2-2, 3-3 and 4-4. Check it by number. Sixes: 0-6, 1-6, 2-6, 4-6, 5-6 and 6-6 are on the table and the 3-6 is in your hand, seven of seven, so no opponent holds a 6. Fives: 0-5, 1-5, 2-5, 3-5, 4-5, 5-5 and 5-6 are all on the table, so 5 is dead as well. Threes: 2-3, 3-4 and 3-5 on the table, 3-6 and 0-3 in your hand, leaving two unseen, the 1-3 and 3-3. Zeros: five on the table (1-0, 0-2, 5-0, 0-0, 0-6), one in your hand, one unseen, the 0-4. The six unseen tiles weigh 4 + 4 + 5 + 4 + 6 + 8 = 31 pips against your 12.

You have three legal plays. Play the 3-6 on the left, on the 3, and both ends show 6. Every 6 is now on the table, so Left, Top and Right must all pass, and so must you, because the 0-3 has no 6. The hand is blocked on the spot. You hold 3 pips. The lightest unseen tile is 4 pips (the 0-4 or the 2-2), so Left, with one tile, holds at least 4; Top, with two, at least 4 + 4 = 8; Right, with three, at least 4 + 4 + 5 = 13. Whatever the distribution, your 3 is the lowest total. You win the blocked hand and score every unseen pip: 31 points, known before you play.

Play the 3-6 on the right instead, on the 6, and the ends show 3 and 3, with the 0-3 ready to go out on either. Only the 1-3 and 3-3 carry a 3 among the unseen tiles, and only the 1-3 can turn a 3 end into something else, so a 3 will still be showing on your next turn. The trouble is Left. If Left’s single tile is the 1-3 or the 3-3, Left plays it, goes out, and scores your 3 pips plus everything Top and Right hold. If not, you go out next turn, but only for what is still in their hands after Top and Right have shed. You would be trading a certain 31 for at most 31, with a loss attached that is decided by a tile you cannot see.

The third play, the 0-3 on the left, keeps the 9-pip 3-6 and leaves the ends at 0 and 6. The 6 is dead and only the 0-4 can answer the 0. A 3 or a 6 can reappear on an end only if the 0-4, then the 1-4, then the 1-3 arrive in that order, and meanwhile you hold 9 pips while three opponents shed theirs. If the line locks, a single tile of 4, 5, 6 or 8 pips beats your 9. It is the worst of the three.

Practising the count

The game here deals four seats on screens 640 pixels or wider and two below that, so both layouts above come up. Play against the Hard opponent, and once you are down to three tiles, write the unseen list before every play: how many tiles, which tiles, and the lightest one. The score awarded when the hand ends is the pip total of the losing hands, so you can check your count against it every time.