1. **Stating the problem:** We have a 3x3 grid of numbers with the last cell missing: $$\begin{matrix} 3 & 7 & 12 \\ 23 & 26 & 30 \\ 48 & 53 & ? \end{matrix}$$ We need to find the missing number represented by $?$. 2. **Observing the pattern:** Let's analyze the rows and columns to find a relationship. 3. **Check row sums:** - Row 1 sum: $3 + 7 + 12 = 22$ - Row 2 sum: $23 + 26 + 30 = 79$ - Row 3 sum: $48 + 53 + ? = 101 + ?$ No clear pattern in sums. 4. **Check column differences:** - Column 1: $3, 23, 48$; differences: $23 - 3 = 20$, $48 - 23 = 25$ - Column 2: $7, 26, 53$; differences: $26 - 7 = 19$, $53 - 26 = 27$ - Column 3: $12, 30, ?$; differences: $30 - 12 = 18$, $? - 30 = x$ The differences in columns increase roughly by 5, 8, and 12 respectively, no clear pattern. 5. **Check row differences:** - Row 1: $7 - 3 = 4$, $12 - 7 = 5$ - Row 2: $26 - 23 = 3$, $30 - 26 = 4$ - Row 3: $53 - 48 = 5$, $? - 53 = y$ No consistent pattern. 6. **Try sum of first two numbers in each row equals the third:** - Row 1: $3 + 7 = 10$, but third is $12$ - Row 2: $23 + 26 = 49$, but third is $30$ - Row 3: $48 + 53 = 101$, so $?$ should be close to $101$ if pattern holds, but no match. 7. **Try sum of first and third equals second:** - Row 1: $3 + 12 = 15$, second is $7$ - Row 2: $23 + 30 = 53$, second is $26$ - Row 3: $48 + ? = 53$, second is $53$ No match. 8. **Try sum of columns:** - Column 1: $3 + 23 + 48 = 74$ - Column 2: $7 + 26 + 53 = 86$ - Column 3: $12 + 30 + ? = 42 + ?$ No clear pattern. 9. **Try differences between rows:** - Row 2 - Row 1: $(23-3, 26-7, 30-12) = (20, 19, 18)$ - Row 3 - Row 2: $(48-23, 53-26, ? - 30) = (25, 27, ? - 30)$ Notice the first two differences in the second set are larger than the first set by 5 and 8 respectively. 10. **Assuming the difference in the third column increases by 11 (following 20,19,18 then 25,27, x):** - The differences in the third column are $18$ then $? - 30$ - If the increase is by 11, then $? - 30 = 18 + 11 = 29$ - So $? = 30 + 29 = 59$ 11. **Check if 59 is among the options:** Yes, option D is 59. **Final answer:** $\boxed{59}$