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In the following 3×3 grid, the letters p, q, s, and t represent unknown numbers:
| 40 | q | 45 |
| 41 | 43 | s |
| p | 44 | t |
The grid must follow these rules:
1. The sum of the numbers in each of the three rows is the same.
2. The sum of the numbers in each of the three columns is the same.
3. The sum of any row does not equal the sum of any column.
Determine the values of p, q, s, and t.
Let’s solve this step by step:
1. Let’s call the sum of each row R and the sum of each column C.
2. From the first row: 40 + q + 45 = R
From the second row: 41 + 43 + s = R
Therefore: 40 + q + 45 = 41 + 43 + s
q + 85 = 84 + s
q = s – 1
3. From the first column: 40 + 41 + p = C
From the second column: q + 43 + 44 = C
Therefore: 40 + 41 + p = q + 43 + 44
81 + p = q + 87
p = q + 6
4. Now we have equations relating p, q, and s. Let’s express everything in terms of s:
q = s – 1
p = (s – 1) + 6 = s + 5
5. We can now write our grid as:
| 40 | s – 1 | 45 |
| 41 | 43 | s |
| s + 5 | 44 | t |
6. The sum of the third row must equal the sum of the first row:
(s + 5) + 44 + t = 40 + (s – 1) + 45
s + 49 + t = s + 84
t = 35
7. Now we know the sum of each row: 40 + (s – 1) + 45 = 84 + s
8. The sum of each column must equal this. Let’s use the first column:
40 + 41 + (s + 5) = 84 + s
86 + s = 84 + s
86 = 84
This is true when s = 42
9. Now we can calculate all values:
s = 42
q = s – 1 = 41
p = s + 5 = 47
t = 35 (as calculated earlier)
10. We can verify that each row sums to 126 and each column sums to 128, satisfying all conditions.
Therefore, p = 47, q = 41, s = 42, and t = 35.
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