Some cells of a 5 × 5 grid are shaded. The rows are numbered R1 to R5 from top to bottom and the columns C1 to C5 from left to right; the cell in row i and column j is written (Ri, Cj).
The following is known:
- No two shaded cells share a side (two shaded cells may touch only at a corner).
- The numbers of shaded cells in rows R1, R2, R3, R4 and R5 are 3, 1, 2, 2 and 2 respectively.
- Every column contains the same number of shaded cells.
- The cell (R4, C5) is shaded.
The following is known:
- No two shaded cells share a side (two shaded cells may touch only at a corner).
- The numbers of shaded cells in rows R1, R2, R3, R4 and R5 are 3, 1, 2, 2 and 2 respectively.
- Every column contains the same number of shaded cells.
- The cell (R4, C5) is shaded.
Question 1
Single correct option
In which column is the only shaded cell of row R2?
- A. C2
- B. C3
- C. C4
- D. C1
Show answer and solution
Answer: (A) C2
Deduction: Row 1 needs 3 non-adjacent shaded cells out of 5, so R1 = C1, C3, C5. Ten cells are shaded, so each column has exactly 2. Column 5 is then complete (R1, R4). R2 cannot use C1, C3 or C5 (they touch R1), so its one cell is C2 or C4. R4 already has C5, so its other cell is not C4.
*If R2 = C4:* column 4 needs one more cell, not in R3 (touches R2) or R4, so R5 = C4. Column 2 then needs two non-adjacent cells from R3–R5, i.e. R3 and R5. R3 has C2 and needs a second cell not touching it: only C4 or C5 — both columns are full ✗.
*So R2 = C2.* Column 4 needs two non-adjacent cells from R3–R5 excluding R4 ⇒ R3 and R5. R3's second cell cannot be C2 (touches R2), C3 or C5 ⇒ C1. Column 1 is now full, so R5 = C2, C4. Column 3 still needs one cell ⇒ R4 = C3, C5.
The deduced grid (■ = shaded, · = unshaded):
| Row | C1 | C2 | C3 | C4 | C5 |
|---|---|---|---|---|---|
| R1 | ■ | · | ■ | · | ■ |
| R2 | · | ■ | · | · | · |
| R3 | ■ | · | · | ■ | · |
| R4 | · | · | ■ | · | ■ |
| R5 | · | ■ | · | ■ | · |
*If R2 = C4:* column 4 needs one more cell, not in R3 (touches R2) or R4, so R5 = C4. Column 2 then needs two non-adjacent cells from R3–R5, i.e. R3 and R5. R3 has C2 and needs a second cell not touching it: only C4 or C5 — both columns are full ✗.
*So R2 = C2.* Column 4 needs two non-adjacent cells from R3–R5 excluding R4 ⇒ R3 and R5. R3's second cell cannot be C2 (touches R2), C3 or C5 ⇒ C1. Column 1 is now full, so R5 = C2, C4. Column 3 still needs one cell ⇒ R4 = C3, C5.
The deduced grid (■ = shaded, · = unshaded):
| Row | C1 | C2 | C3 | C4 | C5 |
|---|---|---|---|---|---|
| R1 | ■ | · | ■ | · | ■ |
| R2 | · | ■ | · | · | · |
| R3 | ■ | · | · | ■ | · |
| R4 | · | · | ■ | · | ■ |
| R5 | · | ■ | · | ■ | · |
Question 2
Numerical answer
How many shaded cells lie on the two diagonals of the grid, i.e. the cells (R1, C1), (R2, C2), …, (R5, C5) and (R1, C5), (R2, C4), …, (R5, C1)? (Count the centre cell only once.)
Show answer and solution
Answer: 3
The deduced grid (■ = shaded, · = unshaded):
| Row | C1 | C2 | C3 | C4 | C5 |
|---|---|---|---|---|---|
| R1 | ■ | · | ■ | · | ■ |
| R2 | · | ■ | · | · | · |
| R3 | ■ | · | · | ■ | · |
| R4 | · | · | ■ | · | ■ |
| R5 | · | ■ | · | ■ | · |
Main diagonal: (R1, C1) ■, (R2, C2) ■, (R3, C3) ·, (R4, C4) ·, (R5, C5) · → 2.
Other diagonal: (R1, C5) ■, (R2, C4) ·, (R3, C3) ·, (R4, C2) ·, (R5, C1) · → 1.
Total = 3. The grid itself follows from row 1 being forced to C1, C3, C5 and the column counts of 2 each.
| Row | C1 | C2 | C3 | C4 | C5 |
|---|---|---|---|---|---|
| R1 | ■ | · | ■ | · | ■ |
| R2 | · | ■ | · | · | · |
| R3 | ■ | · | · | ■ | · |
| R4 | · | · | ■ | · | ■ |
| R5 | · | ■ | · | ■ | · |
Main diagonal: (R1, C1) ■, (R2, C2) ■, (R3, C3) ·, (R4, C4) ·, (R5, C5) · → 2.
Other diagonal: (R1, C5) ■, (R2, C4) ·, (R3, C3) ·, (R4, C2) ·, (R5, C1) · → 1.
Total = 3. The grid itself follows from row 1 being forced to C1, C3, C5 and the column counts of 2 each.
Question 3
Numerical answer
Two shaded cells are said to form a 'corner pair' if they touch only at a corner. How many corner pairs are there in the grid?
Show answer and solution
Answer: 8
The deduced grid (■ = shaded, · = unshaded):
| Row | C1 | C2 | C3 | C4 | C5 |
|---|---|---|---|---|---|
| R1 | ■ | · | ■ | · | ■ |
| R2 | · | ■ | · | · | · |
| R3 | ■ | · | · | ■ | · |
| R4 | · | · | ■ | · | ■ |
| R5 | · | ■ | · | ■ | · |
List each shaded cell with the shaded cells diagonally below it:
- (R1, C1)–(R2, C2); (R1, C3)–(R2, C2)
- (R2, C2)–(R3, C1)
- (R3, C4)–(R4, C3); (R3, C4)–(R4, C5)
- (R4, C3)–(R5, C2); (R4, C3)–(R5, C4); (R4, C5)–(R5, C4)
Total = 8.
Tip: scan only downward-diagonal neighbours so that no pair is counted twice.
| Row | C1 | C2 | C3 | C4 | C5 |
|---|---|---|---|---|---|
| R1 | ■ | · | ■ | · | ■ |
| R2 | · | ■ | · | · | · |
| R3 | ■ | · | · | ■ | · |
| R4 | · | · | ■ | · | ■ |
| R5 | · | ■ | · | ■ | · |
List each shaded cell with the shaded cells diagonally below it:
- (R1, C1)–(R2, C2); (R1, C3)–(R2, C2)
- (R2, C2)–(R3, C1)
- (R3, C4)–(R4, C3); (R3, C4)–(R4, C5)
- (R4, C3)–(R5, C2); (R4, C3)–(R5, C4); (R4, C5)–(R5, C4)
Total = 8.
Tip: scan only downward-diagonal neighbours so that no pair is counted twice.
Question 4
Single correct option
Which of the following cells is NOT shaded?
- A. (R5, C4)
- B. (R3, C1)
- C. (R4, C3)
- D. (R3, C5)
Show answer and solution
Answer: (D) (R3, C5)
The deduced grid (■ = shaded, · = unshaded):
| Row | C1 | C2 | C3 | C4 | C5 |
|---|---|---|---|---|---|
| R1 | ■ | · | ■ | · | ■ |
| R2 | · | ■ | · | · | · |
| R3 | ■ | · | · | ■ | · |
| R4 | · | · | ■ | · | ■ |
| R5 | · | ■ | · | ■ | · |
(R3, C1), (R4, C3) and (R5, C4) are all shaded. (R3, C5) cannot be shaded: it shares a side with (R4, C5), which is given as shaded (and column 5 already has its two cells in R1 and R4). Hence the answer is (R3, C5).
| Row | C1 | C2 | C3 | C4 | C5 |
|---|---|---|---|---|---|
| R1 | ■ | · | ■ | · | ■ |
| R2 | · | ■ | · | · | · |
| R3 | ■ | · | · | ■ | · |
| R4 | · | · | ■ | · | ■ |
| R5 | · | ■ | · | ■ | · |
(R3, C1), (R4, C3) and (R5, C4) are all shaded. (R3, C5) cannot be shaded: it shares a side with (R4, C5), which is given as shaded (and column 5 already has its two cells in R1 and R4). Hence the answer is (R3, C5).
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