Which one will replace the question mark ?

Concept

In a number-triangle series, first find the single arithmetic rule that connects the three outer vertices to the centre number, and confirm that the same rule holds for every triangle in the set whose centre is already given, before applying that rule to find the missing centre.

Working

1. Label each triangle's top-left vertex as a, top-right vertex as b, and bottom vertex as c, with centre value n.

2. Test the first triangle (a = 5, b = 3, c = 4, n = 19): a x b + c = 5 x 3 + 4 = 15 + 4 = 19, which matches.

3. Test the third triangle (a = 6, b = 4, c = 5, n = 29): a x b + c = 6 x 4 + 5 = 24 + 5 = 29, which matches, confirming the rule holds across two independent triangles, not just one.

4. Apply the confirmed rule to the triangle with the missing centre (a = 7, b = 5, c = 6): a x b + c = 7 x 5 + 6 = 35 + 6 = 41.

Cross-check

Neither a plain sum of the three vertices (5 + 3 + 4 = 12 and 6 + 4 + 5 = 15) nor a product of all three (5 x 3 x 4 = 60 and 6 x 4 x 5 = 120) matches the given centre values, so multiplying the two top vertices and adding the bottom vertex is the only relationship consistent with both fully-known triangles, confirming that applying it to the third triangle is the right approach for finding the missing number.

So the missing centre value is 41.