🔧 Step-by-Step Answer Walkthrough For Easy Level
The regions at [0,1], [1,1], [1,4], [2,1], [2,4], and [2,5] each consist of a single cell with a sum target, so each cell must equal its target. This gives immediate fixed values at those six coordinates and is the main deduction engine.
With [0,1] forced to sum 6, the 4/6 domino must lie horizontally at [0,2]–[0,1], putting 4 at [0,2]. With [1,1] forced to sum 5, the 5/0 domino lies horizontally at [1,1]–[1,0], putting 0 at [1,0]. The bottom pair is equally forced: [2,5] takes 6 from the 6/5 domino at [2,4]–[2,5], because [2,4] must be its sum 5.
The single-cell sum 4 at [1,4] forces the 4/2 domino vertically at [1,4]–[0,4], placing 2 at [0,4]. The single-cell sum 4 at [2,1] forces the 3/4 domino horizontally at [2,1]–[2,2], placing 3 at [2,2], which also satisfies the greater-than-2 constraint.
All five dominoes are placed: 4/6, 5/0, 6/5, 3/4, and 4/2. Every sum region matches its target, the [2,2] cell is greater than 2, and the remaining empty cells [0,2], [0,4], and [1,0] are correctly filled by their partners.
🔧 Step-by-Step Answer Walkthrough For Medium Level
Focus on the equal pair [1,4]/[2,4]. If [2,4] were the higher pip of the 4/6 domino, [1,4] would also have to be that value, but no remaining domino on [1,3] can supply the other half. Therefore [2,4] must be 4, the 4/6 domino lies horizontally at [2,3]–[2,4] with 6 at [2,3], and [1,4] is forced to 4.
Because [1,4] is 4, the domino covering [1,3]–[1,4] is the 2/4 domino, so [1,3] is 2. Since [1,3] belongs to the four-cell equal region at [1,1],[1,2],[1,3],[2,2], all four cells must be 2.
The 2/1 domino must cover [1,1] and [0,1], giving [0,1]=1 and satisfying less-than-3. The 2/3 domino must cover [1,2] and [0,2], giving [0,2]=3 and satisfying less-than-6. The 2/5 domino covers [2,2] and [3,2], giving [3,2]=5.
Since [3,1] and [3,2] are equal, [3,2]=5 forces [3,1]=5. Then the 4/5 domino covers [2,1]–[3,1], placing 4 at [2,1]. The equal pair [2,0]/[2,1] then forces [2,0]=4, and the 1/4 domino at [1,0]–[2,0] puts 1 at [1,0], satisfying its less-than-3 region.
The seven dominoes are now assigned: 4/6, 2/1, 4/5, 2/3, 1/4, 2/5, and 2/4. Check the equal regions, the less-than singles at [0,1], [0,2], and [1,0], and the empty cell at [2,3] to confirm the grid is consistent.
🔧 Step-by-Step Answer Walkthrough For Hard Level
The one-cell sum-0 at [3,2] forces a 0 there and, via the 0/6 domino, puts a 6 at [2,2]. The one-cell sum-0 at [9,4] forces the 0/4 domino vertically, placing 0 at [9,4] and 4 at [8,4]. The one-cell sum-2 at [3,0] forces the 2/6 domino at [3,0]–[2,0], giving [2,0]=6. The one-cell sum-3 at [5,4] forces the 3/2 domino at [5,4]–[5,3], giving [5,3]=2.
The region [2,0]–[2,3] is an equals constraint. Since [2,0] and [2,2] are both 6, all four cells must be 6. Thus [2,1] and [2,3] are 6. The 6/5 domino at [2,1]–[1,1] gives [1,1]=5, and the 6/3 domino at [2,3]–[1,3] gives [1,3]=3.
The top-left equals region [0,0],[1,0],[1,1] all match [1,1]=5, so the 5/5 domino covers [0,0]–[1,0]. The sum-9 region [0,4],[1,3],[1,4] now has [1,3]=3, so all three are 3; the 3/3 domino covers [0,4]–[1,4]. The sum-4 pair [2,4]–[3,4] takes the 2/2 domino.
The vertical equals stack [6,4],[7,4],[8,4] is forced to 4 by [8,4], so the 4/4 domino covers [6,4]–[7,4]. The greater-than-9 region at [8,2],[9,2] must be 6 plus 6, so the 6/6 domino covers those two. With [5,3] already 2, the sum-2 pair [5,2]–[5,3] forces [5,2]=0; the 0/1 domino then places 1 at [6,2].
The sum-3 region [6,2],[7,2],[7,3] already has [6,2]=1 from the 0/1 domino, so the remaining two cells must sum to 2; the 1/1 domino covers [7,2]–[7,3]. The unequal column [0,6]–[3,6] uses the 0/2 domino at [0,6]–[1,6] and the 1/5 domino at [2,6]–[3,6], yielding distinct values 0,2,1,5.
All fifteen dominoes are placed: 0/4, 3/2, 5/5, 0/1, 6/6, 0/2, 3/3, 2/6, 1/5, 2/2, 6/5, 4/4, 6/3, 1/1, and 0/6. Confirm every equals, sum, greater, and unequal region matches its target, and the long domino chain has no overlaps.
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