1. **Stating the problem:** We have a 3×3 grid where each cell contains expressions involving logarithms base 10 of numbers formed by digits in red and blue boxes. 2. **Understanding the rules:** - Each digit 0-9 is used only once per blue box and once per red box. - The logarithms are base 10. - The grid is arranged so that values increase both across rows and down columns. - The center cell is given as 1. 3. **Key formulas and properties:** - $\log(ab) = \log a + \log b$ - $\log a^n = n \log a$ - Since the center cell is 1, and it is $\log(\text{red}\times\text{blue})$, this implies: $$\log(\text{red} \times \text{blue}) = 1 \implies \text{red} \times \text{blue} = 10^1 = 10$$ 4. **Using the center cell to find red and blue digits:** - The product of the red and blue numbers in the center cell is 10. - Possible pairs (red, blue) are (1,10), (2,5), (5,2), (10,1), but since digits are 0-9, red and blue must be single digits. - So the only valid pairs are (2,5) or (5,2). 5. **Filling the grid:** - The grid increases in size left to right and top to bottom. - The center cell is 1, so the cells to the right and below must be greater than 1. - The top-left cell must be less than 1. 6. **Assigning digits:** - Let red center = 2, blue center = 5. - Then $\log(2 \times 5) = \log 10 = 1$ matches center. 7. **Top-left cell:** - $\log(\text{red} \times \text{blue}) < 1$ means product < 10. - Choose red=1, blue=4, product=4, $\log 4 \approx 0.602$. 8. **Bottom-right cell:** - Must be >1, choose red=5, blue=6, product=30, $\log 30 \approx 1.477$. 9. **Check uniqueness and increasing order:** - Red digits used: 1,2,5 - Blue digits used: 4,5,6 - All digits unique per box. 10. **Summary of the grid:** | | Blue 4 | Blue 5 | Blue 6 | |----------|---------|---------|---------| | Red 1 | $\log(1\times4)=0.602$ | $\log(1\times5)=0.699$ | $\log(1\times6)=0.778$ | | Red 2 | $\log(2\times4)=0.903$ | $\log(2\times5)=1$ (center) | $\log(2\times6)=1.079$ | | Red 5 | $\log(5\times4)=1.301$ | $\log(5\times5)=1.398$ | $\log(5\times6)=1.477$ | This satisfies increasing size across and down, unique digits per box, and center cell equals 1. **Final answer:** The red digits are 1, 2, 5 and the blue digits are 4, 5, 6 arranged as above.