Question

x ÷
3x
8x
12x
6x
2÷
12x
2÷
21-10
www.kenken.com

Explanation:

Step1: Analyze the 4x4 KenKen grid, focus on the "2÷" cages (2-cell division, result 2, so numbers are \(a\) and \(b\) with \(\frac{a}{b}=2\) or \(\frac{b}{a}=2\), possible pairs: (2,1), (4,2), (6,3), (8,4), (12,6)).

Step2: Look at the "3×" cage (2-cell multiplication, product 3, only (3,1) since 3 is prime). So first row, first two cells: let's say first cell is 3, second is 1? Wait, no, "3×" is 2-cell? Wait, the first row: "3×" is a cage, "8×" is next, "12×" is last. Wait, maybe the grid is 4 columns, 4 rows. Let's list rows:

Row 1: [3× cage, 8× cage, empty, 12× cage]

Row 2: [empty, 6× cage, empty, empty]

Row 3: [2÷ cage, 12× cage, empty, empty]

Row 4: [empty, empty, 2÷ cage, empty]

First, "3×" cage (2 cells? Or more? Wait, KenKen cages: the "3×" is a 2-cell cage? Let's assume 2-cell. So 3×: possible numbers (1,3) since 3=1×3.

"8×" cage (2-cell?): 8=1×8, 2×4.

"12×" cage (2-cell?): 12=1×12, 2×6, 3×4.

Now, the "2÷" cage (2-cell): as above, possible pairs.

Let's start with the "3×" cage (row 1, column 1-2? Wait, the first row has "3×" (column 1), "8×" (column 2), then a cell, then "12×" (column 4). Wait, maybe the first row is 4 cells: column 1 (3×), column 2 (8×), column 3 (empty), column 4 (12×). So "3×" is column 1, row 1: a single cell? No, KenKen cages are groups of cells. Wait, the top-left "3×" is a cage that includes row 1, column 1 and maybe row 2, column 1? Wait, the grid lines: the first vertical line, then between column 1 and 2, column 2 and 3, column 3 and 4. Horizontal lines: row 1-2, 2-3, 3-4.

Wait, let's re-express the grid with coordinates (row, column), 1-based:

Row 1:
- (1,1): 3× cage (maybe 2 cells: (1,1) and (2,1))
- (1,2): 8× cage (maybe 2 cells: (1,2) and (2,2))
- (1,3): empty
- (1,4): 12× cage (maybe 2 cells: (1,4) and (2,4))

Row 2:
- (2,1): part of 3× cage
- (2,2): part of 8× cage (6×)
- (2,3): empty
- (2,4): part of 12× cage

Row 3:
- (3,1): 2÷ cage (maybe 2 cells: (3,1) and (4,1))
- (3,2): 12× cage (maybe 2 cells: (3,2) and (4,2))
- (3,3): empty
- (3,4): empty

Row 4:
- (4,1): part of 2÷ cage
- (4,2): part of 12× cage
- (4,3): 2÷ cage (2 cells: (4,3) and (3,3)? No, (4,3) and (4,4)?)
- (4,4): empty

Wait, the "6×" is in (2,2), so 8× cage (row 1,2 and row 2,2) has product 8? No, (1,2) is 8×, (2,2) is 6×. Wait, maybe the "8×" cage is (1,2) and (2,2)? Then 8× and 6×? No, that can't be. Wait, maybe the first row, first cage is "3×" (1 cell), "8×" (1 cell), then a cell, then "12×" (1 cell). No, KenKen uses cages with operations, so a cage is a group of cells with an operation (×, ÷, +, -) and a target number.

Let's assume it's a 4x4 grid, so numbers 1-4 (since 4x4 KenKen uses 1-4). Wait, 4x4: numbers 1-4, each row and column has 1-4.

So possible numbers: 1,2,3,4.

Now, "3×" cage: product 3, with 1-4 numbers. So only (1,3) or (3,1). So cage has two cells? No, 3 is prime, so if it's a 2-cell cage, numbers are 1 and 3.

"8×" cage: product 8, with 1-4. 8=2×4 (since 1×8 is invalid, 8>4). So 2 and 4.

"12×" cage: product 12, with 1-4. 12=3×4 (since 2×6 invalid, 6>4). So 3 and 4.

"6×" cage: product 6, with 1-4. 6=2×3.

"2÷" cage: division, result 2. So possible (2,1) or (4,2) (since 1-4: 2/1=2, 4/2=2, 6/3 invalid, 8/4=2 but 8>4). So (2,1) or (4,2).

"12×" cage (row 3, column 2): product 12, with 1-4. 3×4=12.

Let's start with row 1, column 1 (3× cage): if it's 3, then row 1, column 2 (8× cage) must be 4 (since 8=2×4, but 3 is already in row 1, column 1, so 4 is possible? Wait, row 1: numbers 1-4, unique.

Wait, let's list constraints:

Row 1: cells (1,1), (1,2), (1,3), (1,4). Must have 1,2,3,4.

Cage (1,1): 3× (product 3) → (1,1)=3, (2,1)=1 (since 3×1=3, and column 1 has 3 and 1, so row 2, column 1=1).

Cage (1,2): 8× (product 8) → (1,2)=4, (2,2)=2 (since 4×2=8, and row 1, column 2=4, row 2, column 2=2; check column 2: 4 (row1), 2 (row2), then row3, column2 and row4, column2 (12× cage: product 12, so 3×4, but 4 is in row1, column2, so row3, column2=3, row4, column2=4? Wait, 3×4=12, yes.

Cage (1,4): 12× (product 12) → (1,4)=3, (2,4)=4? No, 3×4=12, but row1, column1=3, so (1,4) can't be 3. Wait, 12=4×3, but 3 is in (1,1), so (1,4)=4, (2,4)=3 (4×3=12). Then row1: (1,1)=3, (1,2)=4, (1,4)=4? No, duplicate. Oops, mistake.

Wait, 12× cage with 1-4: 3×4=12, so cells must be 3 and 4. So (1,4) and (2,4): one is 3, one is 4. But (1,1)=3, so (1,4) can't be 3, so (1,4)=4, (2,4)=3.

Now row1: (1,1)=3, (1,2)=4, (1,4)=4? No, duplicate. So (1,2) can't be 4. Wait, 8× cage: 2×4=8, so (1,2)=2, (2,2)=4 (2×4=8). Then row1, column2=2, row2, column2=4.

Now row1: (1,1)=3, (1,2)=2, so remaining numbers 1 and 4 for (1,3) and (1,4).

Cage (1,4): 12×, so (1,4) and (2,4) must multiply to 12. Possible 3×4 or 4×3. (1,1)=3, so (2,4) can't be 3 (column 1 has 3, row 2, column 1=1 (from 3× cage: 3×1=3)). So (1,4)=4, (2,4)=3 (4×3=12). Then row1, column4=4, so row1, column3=1 (since row1 needs 1,2,3,4: 3,2,1,4).

Now row1: [3,2,1,4]

Row2: (2,1)=1 (from 3× cage: 3×1=3), (2,2)=4 (from 8× cage: 2×4=8), (2,4)=3 (from 12× cage: 4×3=12), so row2, column3=2 (since row2 needs 1,4,2,3: 1,4,2,3).

Row2: [1,4,2,3]

Now column 3: row1=1, row2=2, so row3, column3 and row4, column3: need 3 and 4. The "2÷" cage in row4, column3: (4,3) and (4,4) or (3,3) and (4,3)? Wait, the "2÷" cage is in row4, column3 and row4, column4? No, the cage is labeled "2÷" at (4,3), so two cells: (3,3) and (4,3) or (4,3) and (4,4). Let's check column 3: row1=1, row2=2, so row3, column3 and row4, column3 are 3 and 4. If "2÷" cage is (4,3) and (4,4), then 4/2=2, but 2 is in row2, column3. Wait, "2÷" means the quotient is 2, so one number is twice the other. So possible (2,1), (4,2), (3,1.5) invalid, (1,0.5) invalid. So in 1-4, (2,1) or (4,2).

Row3, column1: "2÷" cage, so (3,1) and (4,1). Let's see column1: row1=3, row2=1, so row3, column1 and row4, column1: 2 and 4 (since 1-4: 3,1,2,4). So "2÷" cage: 4/2=2, so (3,1)=4, (4,1)=2 (4÷2=2) or (3,1)=2, (4,1)=4 (2÷4=0.5 invalid). So (3,1)=4, (4,1)=2.

Row3, column1=4, so row3: [4,?, ?, ?]

Row3, column2: "12×" cage, (3,2) and (4,2). Column2: row1=2, row2=4, so row3, column2 and row4, column2: 1 and 3 (since 1-4: 2,4,1,3). Product 12: 1×12 invalid, 3×4=12 but 4 is in row2, column2. Wait, mistake: column2 has row1=2, row2=4, so remaining numbers 1 and 3. 1×3=3≠12. So my earlier assumption is wrong.

Wait, maybe the grid is 1-6? No, 4x4 is 1-4. Wait, maybe the "8×" cage is 3 cells? No, KenKen 4x4 is 2 or 4 cells? Wait, no, 4x4 KenKen has cages of 1-4 cells, but product 8 with 1-4: only 2×4 (2 cells) or 1×2×4 (3 cells, 1×2×4=8). Ah! Maybe the "8×" cage is 3 cells: (1,2), (2,2), (3,2)? No, (2,2) is 6×. Wait, the "6×" cage is (2,2) and (3,2)? 6×: 2×3=6, so (2,2)=2, (3,2)=3 (2×3=6) or (2,2)=3, (3,2)=2.

Let's restart with 1-4, 4x4, each row/column 1-4.

Cage (1,1): 3× (product 3) → 2 cells: (1,1)=3, (2,1)=1 (3×1=3).

Cage (2,2): 6× (product 6) → 2 cells: (2,2)=2, (3,2)=3 (2×3=6) or (2,2)=3, (3,2)=2.

Cage (1,2): 8× (product 8) → 2 cells: (1,2)=4, (2,2)=2 (4×2=8) → matches (2,2)=2. So (1,2)=4, (2,2)=2, (3,2)=3 (from 6× cage: 2×3=6? No, 6× cage is (2,2) and (3,2), so 2×3=6, yes. So (2,2)=2, (3,2)=3.

Cage (1,4): 12× (product 12) → 2 cells: (1,4)=4, (2,4)=3 (4×3=12) but (1,2)=4, duplicate. So (1,4)=3, (2,4)=4 (3×4=12). Then row1: (1,1)=3, (1,4)=3, duplicate. Oops, 12× cage must be 3×4, but 3 is in (1,1), so (2,4)=3, (1,4)=4 (4×3=12), but (1,2)=4, duplicate. So "8×" cage must be 3 cells: (1,2), (2,2), (3,2)? No, (2,2) is 6×. Wait, maybe the grid is 1-6, 4x4? No, 4x4 is 1-4. I think I made a mistake in cage size. Let's check the original grid:

Top row: "3×" (column 1, row 1), "8×" (column 2, row 1), then a cell, "12×" (column 4, row 1). So row 1 has 4 cells: (1,1)=3×, (1,2)=8×, (1,3)=empty, (1,4)=12×.

Row 2: (2,1)=empty, (2,2)=6×, (2,3)=empty, (2,4)=empty.

Row 3: (3,1)=2÷, (3,2)=12×, (3,3)=empty, (3,4)=empty.

Row 4: (4,1)=empty, (4,2)=empty, (4,3)=2÷, (4,4)=empty.

So cages:

- 3×: (1,1) (single cell? No, KenKen cages have at least 2 cells. So (1,1) and (2,1): 2 cells, product 3 → (1,1)=3, (2,1)=1.

- 8×: (1,2) and (2,2): 2 cells, product 8 → (1,2)=4, (2,2)=2 (4×2=8) or (1,2)=2, (2,2)=4

Answer:

Step1: Analyze the 4x4 KenKen grid, focus on the "2÷" cages (2-cell division, result 2, so numbers are \(a\) and \(b\) with \(\frac{a}{b}=2\) or \(\frac{b}{a}=2\), possible pairs: (2,1), (4,2), (6,3), (8,4), (12,6)).

Step2: Look at the "3×" cage (2-cell multiplication, product 3, only (3,1) since 3 is prime). So first row, first two cells: let's say first cell is 3, second is 1? Wait, no, "3×" is 2-cell? Wait, the first row: "3×" is a cage, "8×" is next, "12×" is last. Wait, maybe the grid is 4 columns, 4 rows. Let's list rows:

Row 1: [3× cage, 8× cage, empty, 12× cage]

Row 2: [empty, 6× cage, empty, empty]

Row 3: [2÷ cage, 12× cage, empty, empty]

Row 4: [empty, empty, 2÷ cage, empty]

First, "3×" cage (2 cells? Or more? Wait, KenKen cages: the "3×" is a 2-cell cage? Let's assume 2-cell. So 3×: possible numbers (1,3) since 3=1×3.

"8×" cage (2-cell?): 8=1×8, 2×4.

"12×" cage (2-cell?): 12=1×12, 2×6, 3×4.

Now, the "2÷" cage (2-cell): as above, possible pairs.

Let's start with the "3×" cage (row 1, column 1-2? Wait, the first row has "3×" (column 1), "8×" (column 2), then a cell, then "12×" (column 4). Wait, maybe the first row is 4 cells: column 1 (3×), column 2 (8×), column 3 (empty), column 4 (12×). So "3×" is column 1, row 1: a single cell? No, KenKen cages are groups of cells. Wait, the top-left "3×" is a cage that includes row 1, column 1 and maybe row 2, column 1? Wait, the grid lines: the first vertical line, then between column 1 and 2, column 2 and 3, column 3 and 4. Horizontal lines: row 1-2, 2-3, 3-4.

Wait, let's re-express the grid with coordinates (row, column), 1-based:

Row 1:

• (1,1): 3× cage (maybe 2 cells: (1,1) and (2,1))
• (1,2): 8× cage (maybe 2 cells: (1,2) and (2,2))
• (1,3): empty
• (1,4): 12× cage (maybe 2 cells: (1,4) and (2,4))

Row 2:

• (2,1): part of 3× cage
• (2,2): part of 8× cage (6×)
• (2,3): empty
• (2,4): part of 12× cage

Row 3:

• (3,1): 2÷ cage (maybe 2 cells: (3,1) and (4,1))
• (3,2): 12× cage (maybe 2 cells: (3,2) and (4,2))
• (3,3): empty
• (3,4): empty

Row 4:

• (4,1): part of 2÷ cage
• (4,2): part of 12× cage
• (4,3): 2÷ cage (2 cells: (4,3) and (3,3)? No, (4,3) and (4,4)?)
• (4,4): empty

Wait, the "6×" is in (2,2), so 8× cage (row 1,2 and row 2,2) has product 8? No, (1,2) is 8×, (2,2) is 6×. Wait, maybe the "8×" cage is (1,2) and (2,2)? Then 8× and 6×? No, that can't be. Wait, maybe the first row, first cage is "3×" (1 cell), "8×" (1 cell), then a cell, then "12×" (1 cell). No, KenKen uses cages with operations, so a cage is a group of cells with an operation (×, ÷, +, -) and a target number.

Let's assume it's a 4x4 grid, so numbers 1-4 (since 4x4 KenKen uses 1-4). Wait, 4x4: numbers 1-4, each row and column has 1-4.

So possible numbers: 1,2,3,4.

Now, "3×" cage: product 3, with 1-4 numbers. So only (1,3) or (3,1). So cage has two cells? No, 3 is prime, so if it's a 2-cell cage, numbers are 1 and 3.

"8×" cage: product 8, with 1-4. 8=2×4 (since 1×8 is invalid, 8>4). So 2 and 4.

"12×" cage: product 12, with 1-4. 12=3×4 (since 2×6 invalid, 6>4). So 3 and 4.

"6×" cage: product 6, with 1-4. 6=2×3.

"2÷" cage: division, result 2. So possible (2,1) or (4,2) (since 1-4: 2/1=2, 4/2=2, 6/3 invalid, 8/4=2 but 8>4). So (2,1) or (4,2).

"12×" cage (row 3, column 2): product 12, with 1-4. 3×4=12.

Let's start with row 1, column 1 (3× cage): if it's 3, then row 1, column 2 (8× cage) must be 4 (since 8=2×4, but 3 is already in row 1, column 1, so 4 is possible? Wait, row 1: numbers 1-4, unique.

Wait, let's list constraints:

Row 1: cells (1,1), (1,2), (1,3), (1,4). Must have 1,2,3,4.

Cage (1,1): 3× (product 3) → (1,1)=3, (2,1)=1 (since 3×1=3, and column 1 has 3 and 1, so row 2, column 1=1).

Cage (1,2): 8× (product 8) → (1,2)=4, (2,2)=2 (since 4×2=8, and row 1, column 2=4, row 2, column 2=2; check column 2: 4 (row1), 2 (row2), then row3, column2 and row4, column2 (12× cage: product 12, so 3×4, but 4 is in row1, column2, so row3, column2=3, row4, column2=4? Wait, 3×4=12, yes.

Cage (1,4): 12× (product 12) → (1,4)=3, (2,4)=4? No, 3×4=12, but row1, column1=3, so (1,4) can't be 3. Wait, 12=4×3, but 3 is in (1,1), so (1,4)=4, (2,4)=3 (4×3=12). Then row1: (1,1)=3, (1,2)=4, (1,4)=4? No, duplicate. Oops, mistake.

Wait, 12× cage with 1-4: 3×4=12, so cells must be 3 and 4. So (1,4) and (2,4): one is 3, one is 4. But (1,1)=3, so (1,4) can't be 3, so (1,4)=4, (2,4)=3.

Now row1: (1,1)=3, (1,2)=4, (1,4)=4? No, duplicate. So (1,2) can't be 4. Wait, 8× cage: 2×4=8, so (1,2)=2, (2,2)=4 (2×4=8). Then row1, column2=2, row2, column2=4.

Now row1: (1,1)=3, (1,2)=2, so remaining numbers 1 and 4 for (1,3) and (1,4).

Cage (1,4): 12×, so (1,4) and (2,4) must multiply to 12. Possible 3×4 or 4×3. (1,1)=3, so (2,4) can't be 3 (column 1 has 3, row 2, column 1=1 (from 3× cage: 3×1=3)). So (1,4)=4, (2,4)=3 (4×3=12). Then row1, column4=4, so row1, column3=1 (since row1 needs 1,2,3,4: 3,2,1,4).

Now row1: [3,2,1,4]

Row2: (2,1)=1 (from 3× cage: 3×1=3), (2,2)=4 (from 8× cage: 2×4=8), (2,4)=3 (from 12× cage: 4×3=12), so row2, column3=2 (since row2 needs 1,4,2,3: 1,4,2,3).

Row2: [1,4,2,3]

Now column 3: row1=1, row2=2, so row3, column3 and row4, column3: need 3 and 4. The "2÷" cage in row4, column3: (4,3) and (4,4) or (3,3) and (4,3)? Wait, the "2÷" cage is in row4, column3 and row4, column4? No, the cage is labeled "2÷" at (4,3), so two cells: (3,3) and (4,3) or (4,3) and (4,4). Let's check column 3: row1=1, row2=2, so row3, column3 and row4, column3 are 3 and 4. If "2÷" cage is (4,3) and (4,4), then 4/2=2, but 2 is in row2, column3. Wait, "2÷" means the quotient is 2, so one number is twice the other. So possible (2,1), (4,2), (3,1.5) invalid, (1,0.5) invalid. So in 1-4, (2,1) or (4,2).

Row3, column1: "2÷" cage, so (3,1) and (4,1). Let's see column1: row1=3, row2=1, so row3, column1 and row4, column1: 2 and 4 (since 1-4: 3,1,2,4). So "2÷" cage: 4/2=2, so (3,1)=4, (4,1)=2 (4÷2=2) or (3,1)=2, (4,1)=4 (2÷4=0.5 invalid). So (3,1)=4, (4,1)=2.

Row3, column1=4, so row3: [4,?, ?, ?]

Row3, column2: "12×" cage, (3,2) and (4,2). Column2: row1=2, row2=4, so row3, column2 and row4, column2: 1 and 3 (since 1-4: 2,4,1,3). Product 12: 1×12 invalid, 3×4=12 but 4 is in row2, column2. Wait, mistake: column2 has row1=2, row2=4, so remaining numbers 1 and 3. 1×3=3≠12. So my earlier assumption is wrong.

Wait, maybe the grid is 1-6? No, 4x4 is 1-4. Wait, maybe the "8×" cage is 3 cells? No, KenKen 4x4 is 2 or 4 cells? Wait, no, 4x4 KenKen has cages of 1-4 cells, but product 8 with 1-4: only 2×4 (2 cells) or 1×2×4 (3 cells, 1×2×4=8). Ah! Maybe the "8×" cage is 3 cells: (1,2), (2,2), (3,2)? No, (2,2) is 6×. Wait, the "6×" cage is (2,2) and (3,2)? 6×: 2×3=6, so (2,2)=2, (3,2)=3 (2×3=6) or (2,2)=3, (3,2)=2.

Let's restart with 1-4, 4x4, each row/column 1-4.

Cage (1,1): 3× (product 3) → 2 cells: (1,1)=3, (2,1)=1 (3×1=3).

Cage (2,2): 6× (product 6) → 2 cells: (2,2)=2, (3,2)=3 (2×3=6) or (2,2)=3, (3,2)=2.

Cage (1,2): 8× (product 8) → 2 cells: (1,2)=4, (2,2)=2 (4×2=8) → matches (2,2)=2. So (1,2)=4, (2,2)=2, (3,2)=3 (from 6× cage: 2×3=6? No, 6× cage is (2,2) and (3,2), so 2×3=6, yes. So (2,2)=2, (3,2)=3.

Cage (1,4): 12× (product 12) → 2 cells: (1,4)=4, (2,4)=3 (4×3=12) but (1,2)=4, duplicate. So (1,4)=3, (2,4)=4 (3×4=12). Then row1: (1,1)=3, (1,4)=3, duplicate. Oops, 12× cage must be 3×4, but 3 is in (1,1), so (2,4)=3, (1,4)=4 (4×3=12), but (1,2)=4, duplicate. So "8×" cage must be 3 cells: (1,2), (2,2), (3,2)? No, (2,2) is 6×. Wait, maybe the grid is 1-6, 4x4? No, 4x4 is 1-4. I think I made a mistake in cage size. Let's check the original grid:

Top row: "3×" (column 1, row 1), "8×" (column 2, row 1), then a cell, "12×" (column 4, row 1). So row 1 has 4 cells: (1,1)=3×, (1,2)=8×, (1,3)=empty, (1,4)=12×.

Row 2: (2,1)=empty, (2,2)=6×, (2,3)=empty, (2,4)=empty.

Row 3: (3,1)=2÷, (3,2)=12×, (3,3)=empty, (3,4)=empty.

Row 4: (4,1)=empty, (4,2)=empty, (4,3)=2÷, (4,4)=empty.

So cages:

• 3×: (1,1) (single cell? No, KenKen cages have at least 2 cells. So (1,1) and (2,1): 2 cells, product 3 → (1,1)=3, (2,1)=1.

• 8×: (1,2) and (2,2): 2 cells, product 8 → (1,2)=4, (2,2)=2 (4×2=8) or (1,2)=2, (2,2)=4