Which of the following statements MUST be true?
I. 10 is placed in a slot in Row 1.
II. 1 is placed in a slot in Row 4.
Answer: D
Grid shape: Row 1 has columns 1–4, Row 2 has columns 2–4, Row 3 has columns 3–4, and Row 4 has column 4.
- 10 must have nothing to its right and nothing above it, so 10 is at R1C4.
- 1 must have nothing to its left and nothing below it. The possible slots are R1C1, R2C2, R3C3 and R4C4. Condition 3 (same row or column as 10) leaves R1C1 or R4C4.
- 2 and 3 are not in Row 1 or Column 4 (condition 4). So they lie in R2C2, R2C3 or R3C3.
- 9 must sit next to 10, so 9 is at R1C3 or R2C4.
Case: 9 at R2C4. 7 and 8 must avoid Row 2 and Column 4, leaving R1C1, R1C2, R1C3 and R3C3.
- 8 cannot be at R3C3, because R2C3 would have to lie between 8 and 9.
- 8 cannot be at R1C1 or R1C2, because R1C3 would then have to be 9.
- So 8 is at R1C3. 7 cannot be at R3C3 (R2C3 would have to lie between 7 and 8), so 7 is in Row 1 too.
- 4 and 6 must then share Row 2 or Row 3. If Row 2 is 4, 6, 9, then 2 and 3 have only R3C3 left. If Row 3 is 4, 6, then R2C3 must exceed 4, so 2 and 3 cannot both fit in Row 2.
Every option fails, so this case is impossible.
Case: 9 at R1C3. 7 and 8 avoid Row 1 and Column 3, so they lie in R2C4 and R3C4, with 8 above 7. 2 and 3 cannot both be in Row 2, because 4 and 6 need a row and Row 1 has only two free slots. So 4 and 6 fill R1C1 and R1C2, and 1 goes to R4C4. R2C3 must lie between R3C3 and 8, and the only number left is 5. Then 2 and 3 fill R2C2 and R3C3 in either order.
| C1 | C2 | C3 | C4 | |
|---|---|---|---|---|
| Row 1 | 4 | 6 | 9 | 10 |
| Row 2 | 2/3 | 5 | 8 | |
| Row 3 | 3/2 | 7 | ||
| Row 4 | 1 |
Checking every arrangement confirms these are the only two solutions.
10 is always at R1C4 and 1 is always at R4C4, so both statements must be true.
Answer: Both I and II