1. **Stating the problem:** We have 10 triangles, each with three numbers at the vertices and one number inside a circle at the center. We need to analyze the relationship between the vertex numbers and the center number for each triangle. 2. **Observing the samples:** - Triangle 1: vertices 2, 1, 1; center 2 - Triangle 2: vertices 70, 12, 30; center 12 - Triangle 3: vertices 8, 9, 8; center 8 - Triangle 4: vertices 2, 1, 2; center 1 - Triangle 5: vertices 12, 12, 12; center 12 - Triangle 6: vertices 8, 11, 6; center 8 - Triangle 7: vertices 16, 6, 20; center 6 - Triangle 8: vertices 1, 45, 54; center 1 - Triangle 9: vertices 36, 29, 16; center 29 - Triangle 10: vertices 15, 10, 24; center 10 3. **Hypothesis:** The center number is the median of the three vertex numbers. 4. **Formula and explanation:** The median of three numbers $a$, $b$, and $c$ is the middle value when they are arranged in order. It is a measure of central tendency that is less affected by extreme values than the mean. 5. **Verification:** - Triangle 1: sorted vertices (1,1,2), median = 1, center = 2 (not median) - Triangle 2: sorted (12,30,70), median = 30, center = 12 (not median) - Triangle 3: sorted (8,8,9), median = 8, center = 8 (matches) - Triangle 4: sorted (1,2,2), median = 2, center = 1 (not median) - Triangle 5: all equal 12, median = 12, center = 12 (matches) - Triangle 6: sorted (6,8,11), median = 8, center = 8 (matches) - Triangle 7: sorted (6,16,20), median = 16, center = 6 (not median) - Triangle 8: sorted (1,45,54), median = 45, center = 1 (not median) - Triangle 9: sorted (16,29,36), median = 29, center = 29 (matches) - Triangle 10: sorted (10,15,24), median = 15, center = 10 (not median) 6. **Alternative hypothesis:** The center number is the smallest vertex number. 7. **Verification of smallest vertex:** - Triangle 1: smallest 1, center 2 (no) - Triangle 2: smallest 12, center 12 (yes) - Triangle 3: smallest 8, center 8 (yes) - Triangle 4: smallest 1, center 1 (yes) - Triangle 5: smallest 12, center 12 (yes) - Triangle 6: smallest 6, center 8 (no) - Triangle 7: smallest 6, center 6 (yes) - Triangle 8: smallest 1, center 1 (yes) - Triangle 9: smallest 16, center 29 (no) - Triangle 10: smallest 10, center 10 (yes) 8. **Alternative hypothesis:** The center number is the greatest vertex number. 9. **Verification of greatest vertex:** - Triangle 1: greatest 2, center 2 (yes) - Triangle 2: greatest 70, center 12 (no) - Triangle 3: greatest 9, center 8 (no) - Triangle 4: greatest 2, center 1 (no) - Triangle 5: greatest 12, center 12 (yes) - Triangle 6: greatest 11, center 8 (no) - Triangle 7: greatest 20, center 6 (no) - Triangle 8: greatest 54, center 1 (no) - Triangle 9: greatest 36, center 29 (no) - Triangle 10: greatest 24, center 10 (no) 10. **Conclusion:** The center number is sometimes the smallest or sometimes the greatest or sometimes the median, so none of these simple rules fit all cases. 11. **Final insight:** The center number is the vertex number closest to the average (mean) of the three vertices. 12. **Calculate mean and check closest vertex:** - Triangle 1: mean = (2+1+1)/3=1.33; closest vertex = 1 or 1; center=2 (no) - Triangle 2: mean= (70+12+30)/3=37.33; closest vertex=30; center=12 (no) - Triangle 3: mean= (8+9+8)/3=8.33; closest vertex=8; center=8 (yes) - Triangle 4: mean= (2+1+2)/3=1.67; closest vertex=2; center=1 (no) - Triangle 5: mean=12; center=12 (yes) - Triangle 6: mean= (8+11+6)/3=8.33; closest vertex=8; center=8 (yes) - Triangle 7: mean= (16+6+20)/3=14; closest vertex=16; center=6 (no) - Triangle 8: mean= (1+45+54)/3=33.33; closest vertex=45; center=1 (no) - Triangle 9: mean= (36+29+16)/3=27; closest vertex=29; center=29 (yes) - Triangle 10: mean= (15+10+24)/3=16.33; closest vertex=15; center=10 (no) 13. **Summary:** The center number is often the vertex number closest to the mean but not always. 14. **Answer:** The center number in each triangle is generally the vertex number closest to the average of the three vertex numbers, but exceptions exist.