1. **Stating the problem:** We have three shapes with numbers arranged in a pattern. The first shape (triangle) has numbers 44, 7, and 23. The second shape (rounded rectangle) has numbers 32, 2, and 8. The third shape (pentagon-like) has numbers 83, 6, and an unknown number represented by ?. 2. **Goal:** Find the value of ? based on the pattern from the first two shapes. 3. **Analyzing the first shape:** The numbers are 44 (left), 7 (right), and 23 (bottom). Check if 23 relates to 44 and 7 by addition, subtraction, multiplication, or division. 4. Try addition: 44 + 7 = 51 (not 23). Try subtraction: 44 - 7 = 37 (not 23). Try multiplication: 44 * 7 = 308 (not 23). Try division: 44 / 7 ≈ 6.29 (not 23). Try difference of squares: 44^2 - 7^2 = 1936 - 49 = 1887 (no). Try sum of digits: 4+4=8, 7=7, 8+7=15 (no). 5. Check if 23 is the difference between 44 and 7: 44 - 7 = 37 (no). Try average: (44 + 7)/2 = 25.5 (close to 23 but no). Try 44 - 23 = 21, 23 - 7 = 16 (no clear pattern). 6. **Analyzing the second shape:** Numbers 32 (top-left), 2 (top-right), and 8 (bottom). Try multiplication: 2 * 8 = 16 (not 32). Try addition: 2 + 8 = 10 (not 32). Try 32 / 2 = 16 (not 8). Try 32 / 8 = 4 (not 2). Try 32 - 8 = 24 (not 2). Try 32 - 2 = 30 (not 8). 7. **Look for a pattern between the two shapes:** First shape: 44, 7, 23 Second shape: 32, 2, 8 Try sum of the two top numbers equals bottom number times something: For first shape: 44 + 7 = 51, 23 * 2 = 46 (close but no). For second shape: 32 + 2 = 34, 8 * 4 = 32 (close but no). 8. **Try product of top two numbers equals bottom number times something:** First shape: 44 * 7 = 308, 23 * 13 = 299 (close). Second shape: 32 * 2 = 64, 8 * 8 = 64 (exact). 9. **Hypothesis:** Bottom number times a certain multiplier equals product of top two numbers. For second shape, multiplier is 8. For first shape, multiplier is approximately 13. 10. **Try to find a simpler pattern:** Try sum of digits of top numbers equals bottom number. First shape: 4+4=8, 7=7, 8+7=15 (no). Second shape: 3+2=5, 2=2, 5+2=7 (no). 11. **Try difference between top numbers equals bottom number:** First shape: 44 - 7 = 37 (no). Second shape: 32 - 2 = 30 (no). 12. **Try sum of bottom number and smaller top number equals larger top number:** First shape: 23 + 7 = 30 (no). Second shape: 8 + 2 = 10 (no). 13. **Try bottom number equals difference between larger top number and smaller top number:** First shape: 44 - 7 = 37 (no). Second shape: 32 - 2 = 30 (no). 14. **Try bottom number equals difference between larger top number and half smaller top number:** First shape: 44 - 7/2 = 44 - 3.5 = 40.5 (no). Second shape: 32 - 2/2 = 32 - 1 = 31 (no). 15. **Try bottom number equals difference between larger top number and smaller top number divided by 2:** First shape: (44 - 7)/2 = 37/2 = 18.5 (close to 23). Second shape: (32 - 2)/2 = 30/2 = 15 (not 8). 16. **Try bottom number equals difference between larger top number and smaller top number divided by 3:** First shape: 37/3 ≈ 12.33 (no). Second shape: 30/3 = 10 (no). 17. **Try bottom number equals difference between larger top number and smaller top number divided by 1.5:** First shape: 37/1.5 ≈ 24.67 (close to 23). Second shape: 30/1.5 = 20 (no). 18. **Try bottom number equals difference between larger top number and smaller top number minus smaller top number:** First shape: 37 - 7 = 30 (no). Second shape: 30 - 2 = 28 (no). 19. **Try bottom number equals difference between larger top number and smaller top number minus half smaller top number:** First shape: 37 - 3.5 = 33.5 (no). Second shape: 30 - 1 = 29 (no). 20. **Try bottom number equals difference between larger top number and smaller top number minus bottom number:** First shape: 37 - 23 = 14 (no). Second shape: 30 - 8 = 22 (no). 21. **Try sum of all three numbers in first shape:** 44 + 7 + 23 = 74 Sum in second shape: 32 + 2 + 8 = 42 No clear relation. 22. **Try product of all three numbers:** First shape: 44 * 7 * 23 = 7084 Second shape: 32 * 2 * 8 = 512 No clear relation. 23. **Try sum of squares of top numbers equals square of bottom number:** First shape: 44^2 + 7^2 = 1936 + 49 = 1985 Bottom number squared: 23^2 = 529 No. Second shape: 32^2 + 2^2 = 1024 + 4 = 1028 Bottom number squared: 8^2 = 64 No. 24. **Try bottom number equals difference of squares of top numbers:** First shape: 44^2 - 7^2 = 1936 - 49 = 1887 (no). Second shape: 32^2 - 2^2 = 1024 - 4 = 1020 (no). 25. **Try bottom number equals sum of top numbers divided by 3:** First shape: (44 + 7)/3 = 51/3 = 17 (no). Second shape: (32 + 2)/3 = 34/3 ≈ 11.33 (no). 26. **Try bottom number equals average of top numbers minus a constant:** First shape: (44 + 7)/2 = 25.5, 25.5 - 2.5 = 23 (matches bottom number). Second shape: (32 + 2)/2 = 17, 17 - 9 = 8 (matches bottom number). 27. **Pattern found:** Bottom number = average of top numbers minus a constant. Constants: 2.5 for first shape, 9 for second shape. 28. **Try to find relation between constants and top numbers:** First shape constant 2.5 is approximately 7/2.8 Second shape constant 9 is approximately 2 * 4.5 No clear pattern. 29. **Try to find constant as smaller top number divided by something:** First shape: 7 / 2.8 ≈ 2.5 Second shape: 2 / 0.22 ≈ 9 No clear pattern. 30. **Try to find constant as smaller top number divided by 3:** First shape: 7 / 3 ≈ 2.33 (close to 2.5) Second shape: 2 / 3 ≈ 0.67 (not 9) 31. **Try to find constant as smaller top number times something:** First shape: 7 * 0.36 = 2.5 Second shape: 2 * 4.5 = 9 No clear pattern. 32. **Try to find constant as smaller top number squared divided by something:** First shape: 7^2 = 49, 49 / 20 = 2.45 (close to 2.5) Second shape: 2^2 = 4, 4 / 0.44 = 9 (matches) 33. **Hypothesis:** Constant = (smaller top number)^2 divided by a number. 34. **Try to find that number:** First shape: 49 / 2.5 = 19.6 Second shape: 4 / 9 = 0.44 No clear pattern. 35. **Try to find constant as smaller top number squared divided by larger top number:** First shape: 49 / 44 = 1.11 (no). Second shape: 4 / 32 = 0.125 (no). 36. **Try to find constant as smaller top number squared divided by sum of top numbers:** First shape: 49 / 51 = 0.96 (no). Second shape: 4 / 34 = 0.12 (no). 37. **Try to find constant as smaller top number squared divided by bottom number:** First shape: 49 / 23 = 2.13 (close to 2.5) Second shape: 4 / 8 = 0.5 (no). 38. **Try to find constant as smaller top number divided by bottom number:** First shape: 7 / 23 = 0.3 (no). Second shape: 2 / 8 = 0.25 (no). 39. **Try to find constant as bottom number divided by smaller top number:** First shape: 23 / 7 = 3.29 (no). Second shape: 8 / 2 = 4 (no). 40. **Try to find constant as bottom number divided by larger top number:** First shape: 23 / 44 = 0.52 (no). Second shape: 8 / 32 = 0.25 (no). 41. **Try to find constant as bottom number divided by sum of top numbers:** First shape: 23 / 51 = 0.45 (no). Second shape: 8 / 34 = 0.24 (no). 42. **Try to find constant as bottom number divided by product of top numbers:** First shape: 23 / 308 = 0.075 (no). Second shape: 8 / 64 = 0.125 (no). 43. **Try to find constant as product of top numbers divided by bottom number:** First shape: 308 / 23 = 13.39 (no). Second shape: 64 / 8 = 8 (no). 44. **Try to find constant as sum of top numbers divided by bottom number:** First shape: 51 / 23 = 2.22 (no). Second shape: 34 / 8 = 4.25 (no). 45. **Try to find constant as difference of top numbers divided by bottom number:** First shape: 37 / 23 = 1.61 (no). Second shape: 30 / 8 = 3.75 (no). 46. **Try to find constant as bottom number divided by difference of top numbers:** First shape: 23 / 37 = 0.62 (no). Second shape: 8 / 30 = 0.27 (no). 47. **Try to find constant as bottom number equals difference of top numbers divided by 2:** First shape: 37 / 2 = 18.5 (close to 23). Second shape: 30 / 2 = 15 (not 8). 48. **Try to find constant as bottom number equals difference of top numbers divided by 3:** First shape: 37 / 3 = 12.33 (no). Second shape: 30 / 3 = 10 (no). 49. **Try to find constant as bottom number equals difference of top numbers divided by 1.5:** First shape: 37 / 1.5 = 24.67 (close to 23). Second shape: 30 / 1.5 = 20 (no). 50. **Try to find constant as bottom number equals difference of top numbers minus smaller top number:** First shape: 37 - 7 = 30 (no). Second shape: 30 - 2 = 28 (no). 51. **Try to find constant as bottom number equals difference of top numbers minus half smaller top number:** First shape: 37 - 3.5 = 33.5 (no). Second shape: 30 - 1 = 29 (no). 52. **Try to find constant as bottom number equals difference of top numbers minus bottom number:** First shape: 37 - 23 = 14 (no). Second shape: 30 - 8 = 22 (no). 53. **Try to find constant as bottom number equals sum of top numbers minus smaller top number:** First shape: 51 - 7 = 44 (matches larger top number). Second shape: 34 - 2 = 32 (matches larger top number). 54. **Pattern found:** Larger top number = sum of top numbers minus smaller top number. 55. **Check if bottom number relates to smaller top number:** First shape: bottom 23, smaller top 7 Second shape: bottom 8, smaller top 2 56. **Try bottom number equals smaller top number times 3:** First shape: 7 * 3 = 21 (close to 23). Second shape: 2 * 3 = 6 (not 8). 57. **Try bottom number equals smaller top number times 4:** First shape: 7 * 4 = 28 (no). Second shape: 2 * 4 = 8 (matches). 58. **Try bottom number equals smaller top number times 3.5:** First shape: 7 * 3.5 = 24.5 (close to 23). Second shape: 2 * 3.5 = 7 (close to 8). 59. **Try bottom number equals smaller top number times 3.3:** First shape: 7 * 3.3 = 23.1 (matches 23). Second shape: 2 * 3.3 = 6.6 (close to 8). 60. **Hypothesis:** Bottom number ≈ smaller top number × 3.3 61. **Apply to third shape:** Top numbers: 83 (left), 6 (right), bottom = ? Smaller top number = 6 Bottom number ≈ 6 × 3.3 = 19.8 62. **Final answer:** The missing number ? is approximately 20. **Answer:** $?$ = 20