๐ง Step-by-Step Answer Walkthrough For Easy Level
Cells [0,0], [0,1], [0,2] must all be equal. The only domino with two identical pips that can cover two of them is [3,3]. Placing it horizontally from [0,1] to [0,2] forces all three cells to 3, because the third cell must match. So [0,1]=3, [0,2]=3, [0,0]=3.
Cell [2,1] needs a sum of 6 alone. With pip values 0โ6, the only way to reach 6 is to put a 6 there. Domino [4,6] is the only one containing a 6. It must cover [2,1], so place it horizontally with 6 at [2,1] and 4 at the adjacent [2,0]. That satisfies the sum.
Cells [1,0] and [2,0] must be equal, and [2,0] is already 4. So [1,0] must be 4. Domino [4,3] has a 4 and a 3. To get the 4 into [1,0] and the 3 into the still-free [0,0] (which we earlier deduced must be 3), place it vertically: 4 at [1,0], 3 at [0,0]. Now the top-left equals region is consistent.
Cells [1,1], [1,2], [2,2] are all equal. The remaining domino with identical pips is [1,1]. Place it horizontally from [1,1] to [1,2], giving both 1, so [2,2] also becomes 1. The less-1 region at [3,2] demands a pip less than 1, so only 0 works. Domino [0,1] can place its 0 there and its 1 in the cell above, [2,2], which needs to be 1. So place it vertically: 0 at [3,2], 1 at [2,2], completing the puzzle.
๐ง Step-by-Step Answer Walkthrough For Medium Level
The cells [1,2] and [1,3] must sum to 10. The only possible pair with pips 0โ6 is 5+5 (since 6+4 would need a 4 that cannot simultaneously satisfy other constraints). Therefore, both cells must be 5. No single domino carries two 5s, so the 5s will come from two different dominoes.
Domino [4,5] holds a 5; place it vertically with the 5 in [1,2] and the 4 in [2,2]. Domino [5,0] also has a 5; place it vertically with the 5 in [1,3] and the 0 in [2,3]. This leaves the โemptyโ cell [2,3] without constraint, and the equals pair [2,1]-[2,2] now has [2,2]=4.
Because [2,1] must equal [2,2], [2,1] is forced to 4. The domino that can supply a 4 to [2,1] while also covering [3,1] is [0,4]. Place it horizontally with 0 at [3,1] and 4 at [2,1]. Now the equals region [3,1]-[3,2] forces [3,2]=0. Cell [3,3] must be greater than 4, so it must be 5 or 6.
Domino [0,6] can cover [3,2] and [3,3]: place it horizontally with 0 at [3,2] (already 0) and 6 at [3,3]. 6 > 4, satisfying the greater constraint. The cell [3,2]โs value is confirmed.
Cells [1,0] and [1,1] must both be less than 5. The remaining dominoes are [2,1] and [3,2]. Place [2,1] horizontally at [1,0]-[1,1] with 2 and 1 (both <5). The sum-5 region [0,3]-[0,4] needs a total of 5; the only remaining domino with a 3 and a 2 is [3,2]. Place it horizontally with 3 at [0,4] and 2 at [0,3] (2+3=5). Everything fits.
๐ง Step-by-Step Answer Walkthrough For Hard Level
Cells [0,7], [1,7], [2,7], [3,7] sum to 4. With four cells, the only feasible small-pip combination is 1, 0, 0, 3. The top two cells are vertically adjacent and can be covered by domino [1,0]: place it vertically with 1 at [0,7] and 0 at [1,7]. The bottom two cells then must sum to 3 (since 4โ1โ0=3). Domino [3,0] provides exactly a 3 and a 0; place it vertically with 0 at [2,7] and 3 at [3,7]. The column is satisfied.
Cell [0,0] has a single sum target of 4, so it must be 4. Cell [1,0] has a less-than-4 constraint, so it must be 0, 1, 2, or 3. Domino [4,0] can place a 4 at [0,0] and a 0 at [1,0]. Place it vertically: [0,0]=4, [1,0]=0. The less is satisfied.
The region [1,2], [2,0], [2,1], [2,2] must all be equal. The cell [3,2] is a greater-than-3, so it must be 4, 5, or 6. Domino [4,6] can cover [3,2] and [2,2] vertically: place it with 4 at [3,2] (satisfying >3) and 6 at [2,2]. Now the entire equals block becomes 6. So [2,0] and [2,1] need 6s โ use [6,6] horizontally there. [1,2] must be 6, and the only remaining domino with a 6 is [0,6]; place it vertically with 0 at [0,2] (which happily satisfies the less-4 there) and 6 at [1,2].
Cells [5,5], [5,6], [6,6] are all equal. Domino [0,0] can set two of these to 0; place it vertically with 0 at [5,6] and 0 at [6,6]. Then [5,5] must also be 0. To cover [5,5] and the less-4 cell [5,4], use domino [0,2] horizontally: 0 at [5,5] and 2 at [5,4] (2 is <4).
Cells [8,1] and [8,2] must sum to 4. The only remaining domino that can do that is [2,2] โ place it horizontally for 2 and 2. Cells [7,5], [7,6], [8,6] are equals; the only unused equal-pip domino is [1,1]. Place it horizontally at [7,5]-[7,6] for two 1s. Then [8,6] must also be 1. To cover [9,6] and [8,6], use [3,1] vertically with 3 at [9,6] and 1 at [8,6].
Cells [9,4], [9,5], [9,6] sum to 11. We already placed 3 at [9,6] in Step 5. So [9,4]+[9,5] need to add to 8. Domino [6,2] gives exactly that: place it horizontally with 6 at [9,4] and 2 at [9,5] (6+2+3=11). The remaining dominoes [5,6] and [4,4] fill the last gaps: [5,6] horizontally at [6,2]-[6,1] with 5 at [6,2] (>4) and 6 at [6,1] (>5); then [4,4] horizontally at [2,4]-[2,5] with two 4s in the equals-4 region.
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