Keyhole Graphs
1. The problem shows four keyhole-shaped graphs with the top parts containing three numbers each and the rectangular bottoms containing some numbers.
2. The top-left graph has numbers 2, 3, 4 in the circle and 75 in the rectangle.
3. The top-right graph has 3, 4, 5 in the circle and 78 in the rectangle.
4. The bottom-left graph has 3, 5, 7 in the circle and is labeled 2 (no rectangle number).
5. The bottom-right graph has 3, 4, 5 in the circle and is labeled 3 (no rectangle number).
6. We observe the relationship between the numbers in the circle and the rectangle for the top two graphs.
7. For the top-left graph, let's test if multiplication of the three numbers equals 75: $$2 \times 3 \times 4 = 24$$ which is not 75.
8. Try sum: $$2 + 3 + 4 = 9$$ not 75.
9. Try product times sum: $$24 \times 9 = 216$$ not 75.
10. Try sum of squares: $$2^2 + 3^2 + 4^2 = 4 + 9 + 16 = 29$$ not 75.
11. Try product plus sum: $$24 + 9 = 33$$ not 75.
12. Try another approach: take the product of these numbers minus the sum: $$24 - 9 = 15$$ no.
13. Since 75 is given, check factors of 75: $$75 = 3 \times 5 \times 5$$, and 2,3,4 do not match factors.
14. Try sum of squares times 3: $$29 \times 3 = 87$$ too high.
15. Now look at the top-right graph: 3,4,5 and 78.
16. Sum: $$3 + 4 + 5 = 12$$, product: $$3 \times 4 \times 5 = 60$$
17. Sum plus product: $$12 + 60 = 72$$ close to 78.
18. Sum plus product plus 6: $$12 + 60 + 6 = 78$$ matches.
19. Try same for top-left: $$9 + 24 + 6 = 39$$ no.
20. Try sum plus product plus sum of digits: Sum digits 2+3+4=9, product 24, sum 9; 24+9+9=42 no.
21. Another attempt: Top-left rectangle 75, which is 25 times 3.
22. Try taking product times one of the numbers: 24 times 3 equals 72 close to 75.
23. Try product plus sum plus product of first and last number: product = 24, sum=9, 2 x 4 = 8, total 24+9+8=41 no.
24. Try adding squares and product: 29 + 24 = 53 no.
25. Let's try difference between squares: 4^2 - 3^2 - 2^2 = 16 - 9 - 4 = 3 no.
26. Now look at the 4 graphs labeled 1,2,3,4 (top-left top-right bottom-left bottom-right).
27. The bottom ones have no rectangle number but are labeled 2 and 3.
28. Hypothesize the rectangle number is related to the sum of numbers times the label.
29. For the top-left (label 1): sum is 9, rectangle is 75, 75/9 = 8.333.
30. For the top-right (label 1): sum is 12, rectangle is 78, 78/12 = 6.5.
31. No consistent multiplier.
32. Try sum times sum of digits of label.
33. Or, maybe rectangle = (a*b*c) + (a+b+c) * label
34. For top-left (label 1): 24 + 9*1 = 33 no.
35. For top-right (label 1): 60 + 12*1 = 72 no.
36. Try rectangle number minus product is multiple of sum or label.
37. Top-left: 75 - 24 = 51, 51/9 = 5.67 no.
38. Top-right: 78 -60 =18, 18/12=1.5 no.
39. There is no clear arithmetic pattern; possibly the rectangle number is the sum of cubes?
40. Sum of cubes top-left: $$2^3 + 3^3 + 4^3 = 8 + 27 + 64 = 99$$ no.
41. Sum of cubes top-right: $$3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216$$ no.
42. Now, try sum of products of pairs:
$$2\times3 + 3\times4 + 2\times4=6 + 12 + 8=26$$ no.
43. For top-right $$3\times4 + 4\times5 + 3\times5 = 12 + 20 + 15 = 47$$ no.
44. Alternatively, rectangles might be sum of squares plus something.
45. Given insufficient info, conclude that the bottom rectangle numbers correspond to unknown derivations from the three numbers in the circles.
46. The problem likely asks for the rectangle numbers in bottom graphs based on top graphs.
47. The only plausible pattern: rectangle is product of numbers plus sum.
48. Check top-left: $$24 + 9 = 33$$ but rectangle is 75 no.
49. Try product times average: top-left average is $$\frac{2+3+4}{3} = 3$$, product 24, product*average = $$24 \times 3 = 72$$ near 75.
50. Top-right: average is $$\frac{3+4+5}{3}=4$$, product is 60, product*average = $$60 \times 4 = 240$$ no.
51. Try product times max number: top-left max 4, product 24, product x max = 24x4=96 no.
52. Perhaps bottom rectangles are sums of squares times some factor.
53. Without clear formula, final conclusion:
The problem needs more information or clarification to solve.
No definite numeric answers can be confidently computed with the given data.