🔧 Step-by-Step Answer Walkthrough For Easy Level
The four cells in column 2 — (0,2), (1,2), (2,2), (3,2) — must all show the same pip value. Two of those cells are adjacent: (1,2) and (2,2). For two touching cells to hold equal values, a double is the only domino that works. Today's set has exactly one double: [3|3]. Place it vertically at (1,2) and (2,2). Both cells show 3, so the entire equals region must be 3. That means (0,2) and (3,2) also need 3.
The [2|3] domino covers (0,1) and (0,2). Since (0,2) must be 3, the 2-pip goes to (0,1). Similarly, the [5|3] domino covers (3,1) and (3,2). Since (3,2) must be 3, the 5-pip goes to (3,1). Both dominoes orient without ambiguity.
The two-cell region at (3,0) and (3,1) must total 10. Cell (3,1)=5 is already fixed. So (3,0) must be 5 as well. The [5|0] domino covers (3,0) and (2,0) — with (3,0)=5, the blank lands at (2,0).
The only remaining domino is [6|1], which fills (0,0) and (1,0). The unequal region contains (0,0), (0,1)=2, (1,0), and (2,0)=0. Placing [6|1] with 6 at (0,0) and 1 at (1,0) gives the set {6, 2, 1, 0} — all four values distinct. Constraint satisfied. Puzzle complete.
🔧 Step-by-Step Answer Walkthrough For Medium Level
Region (0,3) is a single cell with sum=4, so it must hold exactly 4 pips. Region (3,3) is also a single cell with sum=3, so it must hold exactly 3 pips. Scan the domino list for a tile with a 4-pip face that can reach (0,3): [1|4] placed horizontally covers (0,2)=1 and (0,3)=4. For (3,3)=3: [3|1] placed horizontally covers (3,3)=3 and (3,2)=1.
Cells (2,2) and (3,2) must sum to 1. Cell (3,2)=1 is already set, so (2,2) must be 0. The [0|2] domino connects (2,2) and (1,2) vertically: (2,2)=0, (1,2)=2. Now check sum-5 at (0,2)+(1,1)+(1,2)=5. Cell (0,2)=1 and (1,2)=2 are known, so (1,1)=5-1-2=2. The [3|2] domino connects (2,1) and (1,1) vertically: (2,1)=3, (1,1)=2.
Sum-6 at (2,1)+(3,1)=6: (2,1)=3, so (3,1)=3. The [5|3] domino covers (3,0) and (3,1) horizontally: (3,0)=5, (3,1)=3. Sum-9 at (2,0)+(3,0)=9: (3,0)=5, so (2,0)=4. The [5|4] domino covers (1,0) and (2,0) vertically: (1,0)=5, (2,0)=4.
Cells (1,3) and (2,3) must sum to 12. Among the remaining dominoes, only [6|6] can produce a sum of 12 across two cells. Place it vertically: (1,3)=6, (2,3)=6.
The final domino is [6|5], covering (0,0) and (0,1) horizontally. Sum-16 at (0,0)+(0,1)+(1,0)=16: (1,0)=5 is already set, so (0,0)+(0,1) must equal 11. [6|5] gives 6+5=11 exactly. Place it as (0,0)=6, (0,1)=5. All constraints satisfied. Puzzle complete.
🔧 Step-by-Step Answer Walkthrough For Hard Level
Two regions require zero sums: the equals region at (4,2), (5,1), (5,2), (6,1) forces all four cells to the same value — and the only value reachable here is 0. The sum-0 region at (6,4) and (7,4) forces both to 0 as well. That's six zero-constrained cells across two regions. Today's set contains exactly the dominoes needed to supply them: the [0|0] double plus several other zero-bearing tiles. None of those dominoes can go anywhere else.
Cells (5,1) and (6,1) are adjacent vertically and both must be 0. The [0|0] double is the only domino that places equal values in two touching cells, and 0 is the only value all four equals-region cells can share. Place [0|0] vertically: (5,1)=0, (6,1)=0. The other two cells in the region — (4,2) and (5,2) — must also be 0, each supplied by a different domino.
Cell (4,2) is part of the equals region (must be 0) and connects to (4,3) via the [2|0] domino. With (4,2)=0, [2|0] placed horizontally gives (4,3)=2. Cell (5,2) must also be 0 and sits next to (5,3). The [0|1] domino placed horizontally gives (5,2)=0, (5,3)=1.
Cells (6,4) and (7,4) must sum to 0, so both are 0. The [3|0] domino connects (5,4) and (6,4) vertically: (6,4)=0 and (5,4)=3. The [4|0] domino connects (8,4) and (7,4) vertically: (7,4)=0 and (8,4)=4. Check sum-7 at (8,4)+(9,4)=7: (8,4)=4, so (9,4) must be 3.
Cell (1,0) must be less than 3, meaning it holds 0, 1, or 2. The [2|3] domino covers (1,0) and (0,0). Its faces are 2 and 3 — only 2 satisfies the constraint. Place [2|3] vertically: (1,0)=2, (0,0)=3. (0,0) is an empty-type cell, so no further constraint applies.
Cells (0,4) and (1,4) must be equal. The [6|6] double is the only remaining double in the set — place it vertically: (0,4)=6, (1,4)=6.
Cell (0,8) must equal 4. The [1|4] domino covers (1,8) and (0,8) vertically: (0,8)=4, (1,8)=1. (1,8) is an empty-type cell — no constraint to satisfy.
The region at (1,2) and (2,2) must exceed 8. The [5|4] domino covers (2,2) and (1,2) vertically: (2,2)=5, (1,2)=4. Sum=9>8 ✓. The region at (1,6) and (2,6) must exceed 9. The [4|6] domino covers (1,6) and (2,6) vertically: (1,6)=4, (2,6)=6. Sum=10>9 ✓.
Cell (3,4) must equal 1. The [6|1] domino covers (4,4) and (3,4) vertically: (3,4)=1, (4,4)=6. Now check the 6-cell unequal region at (4,3), (4,4), (4,5), (5,3), (5,4), (5,5) — all six must differ. Known so far: (4,3)=2, (4,4)=6, (5,3)=1, (5,4)=3. Remaining cells are (4,5) and (5,5), which must take values from the unused set. The [5|5] double covers (4,5) and (4,6): with sum=5 required at (4,6), (4,6)=5 and (4,5)=5. The [4|2] domino covers (5,5) and (5,6): (5,5)=4, (5,6)=2. Unequal check: {2, 6, 5, 1, 3, 4} — all six distinct. ✓
Cells (5,6) and (5,7) must be equal. (5,6)=2 is already set, so (5,7)=2. The [1|2] domino covers (6,7) and (5,7) vertically: (6,7)=1, (5,7)=2 ✓. Sum=1 at (6,7)=1 ✓.
Cells (8,2), (9,2), and (9,3) must sum to 3. The [1|1] double covers (8,2) and (9,2) vertically: both show 1. The [3|1] domino covers (9,4) and (9,3) horizontally: (9,4)=3 ✓ (from step 4), (9,3)=1. Sum check: 1+1+1=3 ✓. Sum-7 at (8,4)+(9,4): 4+3=7 ✓. All constraints satisfied. Puzzle complete.
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