Which number replaces the question mark ?

Concept: In a number-triangle series, the centre number follows a fixed arithmetic relationship with the three corner numbers of that triangle. To find a missing corner, first confirm the rule from the complete triangles, then apply the identical rule to the incomplete one.

Application:

1. Triangle 1: corners are 1, 3 and 4. Sum = 1 + 3 + 4 = 8. Half of the sum = 8 ÷ 2 = 4, which matches the given centre number (4). So the rule is: centre = (sum of the three corner numbers) ÷ 2.

2. Triangle 2: corners are 7, 7 and 6. Sum = 7 + 7 + 6 = 20. Half of the sum = 20 ÷ 2 = 10, which matches the given centre number (10), confirming the rule holds across triangles.

3. Triangle 3: corners are 3, 5 and the missing number x, with centre 6. Applying the same rule: (3 + 5 + x) ÷ 2 = 6.

4. Multiply both sides by 2: 3 + 5 + x = 12, i.e. 8 + x = 12.

5. Solve for x: x = 12 − 8 = 4.

Cross-check: Substitute x = 4 back into the triangle: (3 + 5 + 4) ÷ 2 = 12 ÷ 2 = 6, which exactly matches the given centre number, confirming the value.

So the missing number is 4.