1. Stating the problem: We have three triangles, each with three numbers at the vertices and one number inside. We need to find the missing number inside the third triangle. 2. Observing the first triangle: vertices = 9, 8, 12; inside = 6. 3. Observing the second triangle: vertices = 5, 6, 15; inside = 2. 4. Hypothesize a relation involving vertices and inside number. Let's examine if the inside number relates to the three vertices by some arithmetic relation. 5. Calculate sum and differences: - For triangle 1: sum of vertices = 9 + 8 + 12 = 29 - For triangle 2: sum of vertices = 5 + 6 + 15 = 26 6. Check the difference between the inside number and the sum or other operations: - Triangle 1: 29 - 6 = 23 - Triangle 2: 26 - 2 = 24 No direct pattern there. 7. Try sum of vertices minus inside number divided by something: Check average of vertices minus inside number: - Triangle 1: average vertices = 29 / 3 ≈ 9.67; 9.67 - 6 = 3.67 - Triangle 2: average vertices = 26 / 3 ≈ 8.67; 8.67 - 2 = 6.67 No simple pattern. 8. Try the product of vertices divided by inside number: - Triangle 1: product = 9 * 8 * 12 = 864; 864 / 6 = 144 - Triangle 2: product = 5 * 6 * 15 = 450; 450 / 2 = 225 No clear relation. 9. Try difference between largest and smallest vertex number minus inside number: - Triangle 1: 12 - 8 = 4; 4 - 6 = -2 - Triangle 2: 15 - 5 = 10; 10 - 2 = 8 No clear pattern. 10. Now check the ratio of product of inside number and middle vertex to sum: - For triangle 1, middle vertex let's say 8: 6 * 8 = 48; 48 / 29 ≈ 1.66 - For triangle 2: 2 * 6 = 12; 12 / 26 ≈ 0.46 No clear pattern. 11. Alternative approach: Try the sum of the two top vertices minus the inside number equals the bottom vertex? - Triangle 1: 9 + 8 - 6 = 11 (bottom vertex = 12) No. - Triangle 2: 5 + 6 - 2 = 9 (bottom vertex=15) No. No pattern. 12. Check if product of two top vertices divided by inside number equals bottom vertex: - Triangle 1: (9*8)/6=72/6=12 (Matches bottom vertex 12) - Triangle 2: (5*6)/2=30/2=15 (Matches bottom vertex 15) This matches the bottom vertex in both triangles. 13. Apply this rule for the third triangle to find inside number: - Triangle 3 vertices: 8 (top-left), 7 (top-right), bottom = 14 - Use formula: (8*7)/ ? = 14 - Calculate ?: ? = (8*7)/14 = 56/14 = 4 14. Final answer: The missing inside number for the third triangle is 4.