1. The problem shows pairs of numbers that combine to form a third number in a box: - Top-left: 2 and 3 combine to 31 - Top-right: 4 and 1 combine to 17 - Bottom-right: 7 and 3 combine to ? 2. Let's look for a pattern relating the two numbers (a,b) to the box value: - For (2,3) = 31 - For (4,1) = 17 3. Check if the box is related to the square of the sum: $$ (2+3)^2 = 5^2 = 25 \ne 31 $$ $$ (4+1)^2 = 5^2 = 25 \ne 17 $$ No direct match. 4. Check if the box equals $a^2 + b^2 + (a\times b)$: For (2,3): $$2^2 + 3^2 + (2\times3) = 4 + 9 + 6 = 19$$ For (4,1): $$4^2 + 1^2 + (4\times1) = 16 + 1 + 4 = 21$$ Does not match. 5. Try $a^3 + b^3$: For (2,3): $$2^3 + 3^3 = 8 + 27 = 35$$ For (4,1): $$64 + 1 = 65$$ No match. 6. Try $a^2 + b^2 + a + b$: For (2,3): $$4 + 9 + 2 + 3 = 18$$ For (4,1): $$16 + 1 + 4 + 1 = 22$$ No match. 7. Explore difference between box and $(a + b)^3$: For (2,3): $(5)^3=125$, box=31 (difference 94) No obvious simple pattern. 8. Consider an alternate pattern: Box equals concatenation of $(a+b)$ and $(a \times b)$: For (2,3): $5$ and $6$ => '56' (not 31) For (4,1): $5$ and $4$ => '54' (not 17) No. 9. Try $a \times 10 + b^2$: For (2,3): $2 \times 10 + 9 = 29$ close but no. 10. Try $a^2 + 10 \times b$: For (2,3): $4 + 30 = 34$ no. 11. Try $a \times 10 + b^3$: For (2,3): $20 + 27 = 47$, no. 12. Check if box values correspond to $a^2 \times b$ plus/minus a constant: For (2,3): $4 \times 3 = 12$, no. 13. Given the lack of clear algebraic pattern and the problem type, we can hypothesize that the value in the box is calculated as $10a + b^2$ plus or minus a constant. Calculate the differences: - For (2,3): $10 \times 2 + 3^2 = 20 + 9 = 29$ difference to 31 is +2. - For (4,1): $10 \times 4 + 1 = 40 + 1 = 41$ difference to 17 is -24, no consistent difference. 14. Consider box value equals $a^3 + b^2$: For (2,3): $8 + 9 = 17$ no. 15. Another way is box value = $a^4 - b^3$: For (2,3): $16 - 27 = -11$ no. 16. Because operations are unclear, another plausible assumption is the box values correspond to the sum of cubes minus sum of inputs: Check for (2,3): $8 + 27 - (2 + 3) = 35 -5 = 30$, close to 31. (4,1): $64 + 1 - (4 + 1) = 65 -5 = 60$, no. 17. Due to no discernible pattern, an inference can be made the value in the box is $(a+b)^3 - (b^2 + a)$: For (2,3): $(5)^3 - (9 + 2) = 125 -11 =114$ no. 18. Since none fit, check if the box is the square of the first number plus the cube of the second minus 4: For (2,3): $4 + 27 -4 = 27$ no. 19. Alternatively, just calculate the last box with the last pair (7,3) using the closest observed pattern: Top-left (2,3) box 31: $2^3 + 3^2 = 8 + 9 =17$, off by 14. Top-right (4,1) box 17: $4^3 + 1^2 = 64 +1 =65$, off by 48. Since difference is not consistent no pattern. 20. Final step: For (7,3), possibility the box is sum of cubes: $$7^3 + 3^3 = 343 + 27 = 370$$ So box value is \boxed{370}.