Each of the numbers 1 to 9 is written in exactly one cell of a 3 × 3 grid. The rows are numbered 1 to 3 from top to bottom and the columns 1 to 3 from left to right. The following information is known:
1. The sums of the numbers in rows 1, 2 and 3 are 13, 15 and 17 respectively.
2. The sums of the numbers in columns 1, 2 and 3 are 15, 13 and 17 respectively.
3. The largest number in the grid is in the centre cell.
4. The number in the top-left corner is greater than the number in the bottom-left corner.
5. The number in the top-right corner is odd.
The main diagonal runs from the top-left cell to the bottom-right cell.
1. The sums of the numbers in rows 1, 2 and 3 are 13, 15 and 17 respectively.
2. The sums of the numbers in columns 1, 2 and 3 are 15, 13 and 17 respectively.
3. The largest number in the grid is in the centre cell.
4. The number in the top-left corner is greater than the number in the bottom-left corner.
5. The number in the top-right corner is odd.
The main diagonal runs from the top-left cell to the bottom-right cell.
Question 1
Numerical answer
What number is in the bottom-right cell?
Show answer and solution
Answer: 8
Centre = 9. Row 2 and column 2 then have 6 and 4 left for their other two cells: row 2 outer cells sum to 6 and column 2 outer cells sum to 4.
Column 2 outer cells sum to 4 from distinct numbers → {1, 3}. Row 2 outer cells sum to 6 → {2, 4} (since 1 and 3 are used, {1,5} and {5,1} fail).
Testing the remaining numbers {5, 6, 7, 8} in the corners against the row, column and corner conditions leaves exactly one grid:
| 7 | 1 | 5 |
|---|---|---|
| 2 | 9 | 4 |
| 6 | 3 | 8 |
Bottom-right = 8.
Column 2 outer cells sum to 4 from distinct numbers → {1, 3}. Row 2 outer cells sum to 6 → {2, 4} (since 1 and 3 are used, {1,5} and {5,1} fail).
Testing the remaining numbers {5, 6, 7, 8} in the corners against the row, column and corner conditions leaves exactly one grid:
| 7 | 1 | 5 |
|---|---|---|
| 2 | 9 | 4 |
| 6 | 3 | 8 |
Bottom-right = 8.
Question 2
Single correct option
What is the sum of the numbers on the main diagonal?
- A. 25
- B. 21
- C. 22
- D. 24
Show answer and solution
Answer: (D) 24
Main diagonal = top-left + centre + bottom-right = 7 + 9 + 8 = 24.
Why the corners are fixed: with 1 and 3 in column 2 (top and bottom middle), row 1's corners sum to 13 − b and row 3's to 17 − h. If b = 1, the top corners sum to 12 → {5, 7}; the bottom corners sum to 14 → {6, 8}. Column sums 15 and 17, together with 'top-left > bottom-left' and 'top-right is odd', then force 7, 5 / 6, 8 and d = 2, f = 4. The alternative b = 3 leads to contradictions in the column sums.
Why the corners are fixed: with 1 and 3 in column 2 (top and bottom middle), row 1's corners sum to 13 − b and row 3's to 17 − h. If b = 1, the top corners sum to 12 → {5, 7}; the bottom corners sum to 14 → {6, 8}. Column sums 15 and 17, together with 'top-left > bottom-left' and 'top-right is odd', then force 7, 5 / 6, 8 and d = 2, f = 4. The alternative b = 3 leads to contradictions in the column sums.
Question 3
Single correct option
In which cell is the number 2 placed?
- A. Row 3, column 2
- B. Row 1, column 2
- C. Row 2, column 1
- D. Row 2, column 3
Show answer and solution
Answer: (C) Row 2, column 1
From the solved grid:
| 7 | 1 | 5 |
|---|---|---|
| 2 | 9 | 4 |
| 6 | 3 | 8 |
The number 2 is in row 2, column 1.
Check column 1: 7 + 2 + 6 = 15 ✓. If 2 and 4 were swapped, column 1 would total 17 and column 3 would total 15, reversing the given column sums.
| 7 | 1 | 5 |
|---|---|---|
| 2 | 9 | 4 |
| 6 | 3 | 8 |
The number 2 is in row 2, column 1.
Check column 1: 7 + 2 + 6 = 15 ✓. If 2 and 4 were swapped, column 1 would total 17 and column 3 would total 15, reversing the given column sums.
Question 4
Numerical answer
What is the product of the four numbers in the corner cells of the grid?
Show answer and solution
Answer: 1680
The corner cells hold 7 (top-left), 5 (top-right), 6 (bottom-left) and 8 (bottom-right).
Product = 7 × 5 × 6 × 8 = 35 × 48 = 1680.
Observation: The four middle-edge cells and the centre use 1, 2, 3, 4 and 9, so the corners must be exactly {5, 6, 7, 8} regardless of order — the product can be found once that is established, even before the exact positions are fixed.
Product = 7 × 5 × 6 × 8 = 35 × 48 = 1680.
Observation: The four middle-edge cells and the centre use 1, 2, 3, 4 and 9, so the corners must be exactly {5, 6, 7, 8} regardless of order — the product can be found once that is established, even before the exact positions are fixed.
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