🔧 Step-by-Step Answer Walkthrough For Easy Level
The four‑cell region at [0,1], [0,2], [0,3], [1,3] must sum to 1. The only way with pip values 0‑6 is to have three 0s and one 1. The domino [0,0] is the only double‑zero available, so it must cover two of those zeros. The only adjacent pair that can both be zero is [0,3] and [1,3] (vertical) — place the [0,0] domino there.
With the two rightmost cells now zero, the remaining cells [0,1] and [0,2] must be one and zero. The [1,0] domino fits horizontally, giving 1 at [0,1] and 0 at [0,2], completing the sum‑1 constraint.
The region at [1,0] and [2,0] requires equal values. [1,0] is currently empty. The only unused domino that can pair with a matching neighbor on [2,0] is [4,5] and [6,4] working together: place the [4,5] domino with the 4 at [1,0] (5 goes to [1,1]), setting [1,0]=4. To make [2,0] also 4, the [6,4] domino must go horizontally on [2,0] (4) and [2,1] (6).
The sum‑7 region covers [2,1] (already 6) and [3,1]. So [3,1] must be 1. The remaining domino [3,1] completes the grid: place it vertically with 1 at [3,1] and 3 at [3,2].
🔧 Step-by-Step Answer Walkthrough For Medium Level
The seven‑cell region at [0,2], [0,3], [0,4], [1,3], [1,4], [2,3], [2,4] demands all different pip values. With seven cells and pips 0‑6, it must use each pip exactly once. This means no double domino can have both ends inside the region unless one value can appear outside — and 3 is the only candidate because [3,3] will place one 3 inside ([0,2]) and one 3 outside ([0,1]).
Put the [3,3] domino horizontally at [0,1]/[0,2], giving a 3 to [0,2] inside the unequal zone. Now look at the remaining six cells in that zone: they must contain 0,1,2,4,5,6. The only domino with a 0 that can fit is [6,0], and 0 cannot repeat, so place it vertically with 0 at [1,4] and 6 at [2,4].
The [2,4] domino now covers [2,3] and [1,3]. To avoid repeating pips already placed, it must give 2 at [2,3] and 4 at [1,3]. The [1,5] domino then naturally fills [0,3] and [0,4] with 1 and 5. The unequal region is fully assigned.
The sum‑11 region [0,6]/[0,7] needs two numbers adding to 11. The only unused domino with a high enough pair is [5,6]. Place it on [0,5] and [0,6] (5 and 6). Then [0,7] must be 5 from the [2,5] domino, whose 2 lands at [1,7]. The right equals region forces [2,7] to also be 2; the [1,2] domino satisfies this with the 2 at [2,7] and its 1 at [3,7] to meet the sum‑1 target.
The left equals region [1,0]/[2,0] forces matching values. Place the [0,4] domino with 0 at [0,0] and 4 at [1,0]. To match, [2,0] must be 4, provided by the [5,4] domino (5 at [3,0] to satisfy the sum‑5 there and 4 at [2,0]). All dominoes are placed.
🔧 Step-by-Step Answer Walkthrough For Hard Level
The six‑cell region at [2,1], [3,1], [3,2], [4,1], [4,2], [4,3] must all share the same pip. Checking the domino list, only pip 1 appears enough times across different dominoes — specifically in [2,1] (one 1), [1,1] (two 1s), [1,3] (one 1), [1,0] (one 1), and [5,1] (one 1). No other pip can cover six cells. So this entire region must be 1.
The sum‑10 region [0,0]/[1,0] needs 5+5. The only dominoes with 5 that can be placed here are [5,3] (5,3) and [5,6] (5,6). Place [5,3] horizontally on [0,0]/[0,1], setting [0,0]=5 and [0,1]=3. Then place [5,6] vertically with 5 at [1,0] and 6 at [2,0], completing sum‑10. The sum‑5 region [0,1]/[1,1] now has [0,1]=3, so [1,1] must be 2. The [2,1] domino provides this: 2 at [1,1] and 1 at [2,1] (adding a 1 to the equals zone).
The region [5,0]/[5,1]/[5,2] must be equal. The only way to get three matching cells from the remaining dominoes is to use [4,4] and [4,6]. Place [4,6] with 4 at [5,0] and 6 at [4,0]; place [4,4] with 4 at [5,1] and 4 at [5,2]. This forces [4,0]=6. The sum‑12 left column ([2,0]=6, [3,0], [4,0]=6) then requires [3,0]=0. The [1,0] domino fits perfectly: 0 at [3,0] and 1 at [3,1] (another 1 for the equals region).
The region [2,2]/[2,3] needs a total of 10, so both cells must be 5. The [5,1] domino places 5 at [2,2] and 1 at [3,2] (equals zone), while the [4,5] domino gives 5 at [2,3] and 4 at [3,3], satisfying the single‑cell sum‑4 at [3,3].
The sum‑6 cell at [4,4] is filled by the [6,6] domino, which also places 6 at [4,5] (giving the first value for the sum‑9 region [4,5]/[5,5]). The sum‑9 now needs 3 at [5,5], so the [3,3] domino goes to [5,4]/[5,5] (both 3). The remaining sum‑6 region [5,3]/[5,4] already has a 3 at [5,4]; place the [1,3] domino with 3 at [5,3] and 1 at [4,3] (the final cell of the equals region).
All dominoes are now placed: [5,6] (index 0) at [1,0]/[2,0]; [1,1] (1) at [4,1]/[4,2]; [4,4] (2) at [5,1]/[5,2]; [5,3] (3) at [0,0]/[0,1]; [2,1] (4) at [1,1]/[2,1]; [4,5] (5) at [3,3]/[2,3]; [1,3] (6) at [4,3]/[5,3]; [4,6] (7) at [5,0]/[4,0]; [3,3] (8) at [5,4]/[5,5]; [1,0] (9) at [3,1]/[3,0]; [6,6] (10) at [4,4]/[4,5]; [5,1] (11) at [2,2]/[3,2]. Every constraint is met.
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