🔧 Step-by-Step Answer Walkthrough For Easy Level
The equals region spanning [1,0], [2,0], [2,1] requires three identical pips. Among the available dominoes, only value 3 appears on enough tiles: the double-3 (domino 4) and the 0-3 (domino 0). This forces the region to be all 3s.
Domino 4 (3|3) must cover two of the equals cells. The geometry allows vertical placement at [1,0]-[2,0], setting both to 3.
The third equals cell [2,1] must get its 3 from domino 0 (0|3), placed horizontally at [2,1]-[2,2]. That puts a 3 at [2,1] and a 0 at [2,2], which lies in the sum-8 region. With 0 fixed, the other two cells [2,3] and [3,3] must sum to 8 — so both must be 4. Domino 3 (4|4) fits vertically there.
The isolated sum-3 cell [0,2] demands a 3, so domino 2 (3|6) goes horizontally [0,2]-[0,3] with 3 at [0,2] and 6 at [0,3]. Then the less-6 cell [0,1] needs a pip under 6; the only remaining place for domino 1 (5|6) is horizontally [0,1]-[0,0] with 5 at [0,1] and 6 at [0,0] (empty).
🔧 Step-by-Step Answer Walkthrough For Medium Level
The equals region occupies [1,0], [2,0], [3,0]. Only value 3 can appear three times (double-3 domino 8, plus 3 from domino 6 or 1). So the column gets 3s. Place domino 8 (3|3) vertically at [2,0]-[3,0]. To give [1,0] a 3, use domino 6 (6|3) vertically at [0,0]-[1,0] with 6 at [0,0] and 3 at [1,0].
[0,0] is 6, so the sum-11 pair [0,0]-[0,1] needs 5 at [0,1]. Domino 0 (5|1) placed horizontally [0,1]-[0,2] gives 5 and 1. Now [0,2]=1 is part of the sum-5 region [0,2]-[0,4]; the other two cells must sum to 4, forcing both 2. Use domino 3 (2|2) horizontally [0,3]-[0,4].
A sum-4 singleton at [0,5] requires a 4. Domino 5 (4|4) covers [0,5]-[0,6] with two 4s, simultaneously satisfying the equals pair [0,6]-[0,7]. So [0,7] becomes 4. Domino 7 (4|5) then fits [0,7]-[1,7] with 4 at [0,7] and 5 at [1,7].
The sum-11 pair [1,7]-[2,7] has 5 at [1,7], so [2,7] must be 6. Place domino 4 (0|6) vertically [2,7]-[3,7] with 6 at [2,7] and 0 at [3,7] (leftover empty cell).
The region [2,3]-[3,4] needs four distinct values. With remaining dominoes 1 (1|3) and 2 (0|2), place domino 1 vertically [2,3]-[3,3] (3 at [2,3], 1 at [3,3]) and domino 2 vertically [2,4]-[3,4] (2 at [2,4], 0 at [3,4]). All values 3,2,1,0 — unique.
🔧 Step-by-Step Answer Walkthrough For Hard Level
The sum-1 cell [3,0] must be 1. The only domino that can place a 1 there while covering upward is domino 12 (5|1), placed vertically [2,0]-[3,0] (5 above, 1 below). Then the sum-11 pair [1,0]-[2,0] now has 5 at [2,0], so [1,0] must be 6. Place domino 6 (3|6) vertically [0,0]-[1,0] (3 at [0,0], 6 at [1,0]).
[0,0] is 3, so the sum-11 region [0,0]-[0,2] needs 8 more across [0,1] and [0,2]. The only domino that fits is 4|4, placed horizontally [0,1]-[0,2] (both 4). The empty corners [0,7]-[1,7] can be filled with domino 10 (0|0) horizontally.
The sum-1 at [3,1] forces a 1 there. Domino 7 (2|1) placed horizontally [3,1]-[3,2] gives 1 at [3,1] and 2 at [3,2] (empty). Then tackle the sum-11 strip in the middle: [1,4]-[2,4] must sum to 11. Place domino 8 (6|6) horizontally [0,4]-[1,4] to satisfy the greater-4 cell at [0,4] and put 6 at both; then [2,4] needs 5. Use domino 5 (5|2) vertically [2,4]-[3,4] (5 at [2,4], 2 at [3,4]).
The sum-6 region [3,4]-[3,5] now has 2 at [3,4], so [3,5] must be 4. Domino 4 (3|4) placed horizontally [3,5]-[3,6] puts 4 at [3,5] and 3 at [3,6]. The equals region [3,6]-[3,7] then demands both 3, so [3,7] must be 3. Place domino 0 (3|0) vertically [2,7]-[3,7] with 3 at [3,7] and 0 at [2,7].
The sum-1 pair [5,3]-[6,3] needs 0 and 1. Domino 3 (5|0) horizontally [5,3]-[5,4] sets [5,3]=0 and [5,4]=5. Then domino 9 (1|1) horizontally [6,3]-[7,3] sets both 1, completing sum-1. The sum-11 region [5,4]-[5,5] gets [5,4]=5, so [5,5] must be 6. Place domino 11 (2|6) vertically [5,5]-[6,5] (6 at [5,5], 2 at [6,5]).
The sum-11 region [7,3]-[9,3] has 1 at [7,3]; the other two cells need sum 10, so use domino 1 (5|5) vertically [8,3]-[9,3] (both 5). Finally, sum-4 region [6,5]-[7,5] has 2 at [6,5], so [7,5] needs 2. The sum-4 singleton [7,4] needs 4. Domino 13 (2|4) horizontally [7,4]-[7,5] places 4 at [7,4] and 2 at [7,5].
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