Question: 2) The table consists of numbers from 1 to 56. Kay and Lin are given a plastic frame that covers exactly 9 squares of the table with the centre square darkened. a. Kay puts the frame on the 9 square with the upper right corner on number 5. What is the average of the numbers that can be seen in the frame? b. Lin puts the frame on some other 9 squares. The sum of the 8 numbers that can be seen in the frame is 248. What is the largest number that can be seen in the frame?

1. **Problem Statement:** We have a table of numbers from $1$ to $56$ arranged in $7$ rows and $8$ columns. A plastic frame covers exactly $9$ squares in a $3 \times 3$ grid with the center square darkened. 2. **Understanding the frame placement:** The frame covers $9$ squares arranged in $3$ rows and $3$ columns. The center square is darkened and is included in the frame. 3. **Part (a): Kay's frame placement** - The upper right corner of the frame is on number $5$. - Since the frame is $3 \times 3$, the squares covered are: - Top row: numbers $3, 4, 5$ - Middle row: numbers $11, 12, 13$ - Bottom row: numbers $19, 20, 21$ 4. **Calculate the sum of these numbers:** $$3 + 4 + 5 + 11 + 12 + 13 + 19 + 20 + 21 = (3+4+5) + (11+12+13) + (19+20+21)$$ $$= 12 + 36 + 60 = 108$$ 5. **Calculate the average:** The frame covers $9$ squares, so the average is: $$\frac{108}{9} = 12$$ 6. **Answer for (a):** The average of the numbers in the frame is $12$. 7. **Part (b): Lin's frame placement** - The sum of the $8$ visible numbers (excluding the darkened center) is $248$. - We want to find the largest number that can be seen in the frame. 8. **Let the center number be $x$.** - The total sum of all $9$ numbers in the frame is $248 + x$. 9. **Properties of the frame:** - The frame covers a $3 \times 3$ block in the $7 \times 8$ table. - The numbers increase by $1$ from left to right and top to bottom. 10. **Express the numbers in the frame:** Let the top-left corner of the frame be at position with number $n$. Then the frame numbers are: $$n, n+1, n+2$$ $$n+8, x, n+10$$ $$n+16, n+17, n+18$$ where $x = n+9$ is the center number. 11. **Sum of all 9 numbers:** $$S = n + (n+1) + (n+2) + (n+8) + (n+9) + (n+10) + (n+16) + (n+17) + (n+18)$$ $$= 9n + (1+2+8+9+10+16+17+18)$$ $$= 9n + 81$$ 12. **Sum of 8 visible numbers (excluding center $x = n+9$):** $$S - x = (9n + 81) - (n + 9) = 8n + 72$$ 13. **Given sum of 8 visible numbers:** $$8n + 72 = 248$$ 14. **Solve for $n$:** $$8n = 248 - 72 = 176$$ $$n = \frac{176}{8} = 22$$ 15. **Find the largest number in the frame:** The largest number is $n + 18 = 22 + 18 = 40$. 16. **Answer for (b):** The largest number that can be seen in the frame is $40$. **Final answers:** - (a) Average is $12$. - (b) Largest number visible is $40$.