1. The problem gives us pairs of numbers that combine into a result. We need to find how the two numbers combine to produce the number in the box. 2. For the top-left graph: two numbers 2 and 3 combine to give 31. 3. For the top-right graph: two numbers 4 and 1 combine to give 17. 4. Let's hypothesize a formula for combining the two numbers $a$ and $b$. Let's denote the output as $f(a,b)$. 5. Try a linear combination. Check if $f(a,b) = 10a + b + c$ for some $c$. 6. Using (2,3) -> 31: $10 \cdot 2 + 3 + c = 31 \Rightarrow 20 + 3 + c = 31 \Rightarrow c = 31 - 23 = 8$. 7. Using (4,1) -> 17: $10 \cdot 4 + 1 + c = 17 \Rightarrow 40 + 1 + c =17 \Rightarrow c = 17 - 41 = -24$. 8. Since $c$ is inconsistent, try another approach. 9. Try $f(a,b) = a^2 + b^2 + ?$ 10. Check (2,3) -> $2^2 + 3^2 = 4 + 9 = 13$ (not 31), (4,1)-> $16 + 1=17$ (matches 17) 11. So this fits (4,1) but not (2,3). 12. Try $f(a,b) = (a+b)^2 - 2ab$ or other variations. 13. Let's check sum multiplied by one input: $f(a,b) = a \times (a+b)$. 14. For (2,3): $2 \times 5 = 10$ (not 31), no. 15. Try $f(a,b) = a^3 + b^3$ 16. (2,3): $8 + 27=35$ (not 31) 17. Try $f(a,b) = (a+b)^3 - (a^3 + b^3)$ 18. (2,3): sum is 5, sum cubed is 125. $125 - (8+27)=125-35=90$ (no) 19. Check difference between output and sum of squares. 20. For (2,3) output 31; sum squares 13; difference 18. 21. For (4,1) output 17; sum squares 17; difference 0. 22. Try $f(a,b) = a^3 + b + k$ for some $k$ 23. (2,3): $8 + 3 + k=31$ gives $k=20$ 24. (4,1): $64 + 1 + k=17$ gives $k= -48$ inconsistent. 25. Try sum squared plus product: $f(a,b) = (a+b)^2 + ab$ 26. (2,3): $5^2 + 6 = 25 + 6 = 31$ matching 31 27. (4,1): $5^2 + 4 = 25 + 4 = 29$ not 17 28. Try sum squared minus product: $f(a,b) = (a+b)^2 - ab$ 29. (2,3): $25 - 6 = 19$ no 30. Try sum squared minus twice product: $f(a,b) = (a+b)^2 - 2ab$ 31. (2,3): $25 - 12 = 13$ no 32. Try product plus sum squared: $ab + (a+b)^2$ 33. (2,3): $6 + 25 = 31$ correct 34. (4,1): $4 + 25 = 29$ no 35. Try $f(a,b) = ab + a^2 + b^2$ 36. (2,3): $6 + 4 + 9 = 19$ no 37. Try different combination for (4,1)=17: $4 imes 4 + 1 = 17$ 38. So (4,1) -> $a^2 + b$ matches 39. Check (2,3) with $a^2 + b$: $2^2 + 3 = 4 + 3 = 7$ no 40. Check (2,3) with $a + b^2$: $2 + 9=11$ no 41. Try $2a^2 + b$: (2,3) $2 imes 4 + 3=11$ no; (4,1) $2 imes 16 + 1=33$ no 42. Check if (2,3): $10a + b + ab=20 +3 + 6=29$ no 43. (4,1): $40 + 1 + 4=45$ no 44. From the above, the pattern matches $f(a,b) = (a+b)^2 + ab$ for (2,3)=31, but not the second case. 45. Conversely, (4,1) matches $a^2 + 2b + 9$? $16 + 2 + 9=27$ no 46. Since no universal pattern, maybe the output is $a^3 + b^2$: (2,3): $8+9=17$ no; (4,1): $64+1=65$ no 47. Focus on applying $f(a,b) = a^2 + b^3$: (2,3): $4 + 27=31$ matches (4,1): $16 + 1 = 17$ matches 48. This fits both given mappings! 49. For bottom-right graph with (7,3), calculate $7^2 + 3^3 = 49 + 27 = 76$. 50. Final answer for the unknown box is 76. **Answer:** 76