Question: 7 (a) The diagram shows part of a number grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 A cross outlining five numbers, as shown, can be placed anywhere on the grid. (i) If $n$ represents the number in the top left corner of the cross, write down an expression, in terms of $n$, for the number in the bottom right corner of the cross. (ii) Show that the difference between the products of the numbers in the opposite corners of the cross is always 36. (iii) Show that the sum of the five numbers in the cross cannot be 1715.

1. **Problem statement:** We have a number grid with rows of 9 numbers each. A cross shape covers five numbers: the top left corner number $n$, the top right corner, the center, the bottom left corner, and the bottom right corner. 2. **Understanding the grid and cross:** Each row has 9 numbers, so moving one row down adds 9 to the number. 3. **(i) Expression for bottom right corner:** - The top left corner is $n$. - The cross covers a 3x3 block with the center at $n + 10$ (since moving one row down and one column right adds $9 + 1 = 10$). - The bottom right corner is two rows down and two columns right from $n$, so it is $n + 2 \times 9 + 2 = n + 18 + 2 = n + 20$. **Answer:** The bottom right corner number is $n + 20$. 4. **(ii) Difference between products of opposite corners:** - Opposite corners are: - Top left: $n$ - Bottom right: $n + 20$ - Top right: $n + 2$ - Bottom left: $n + 18$ - Calculate the difference: $$ (n)(n + 20) - (n + 2)(n + 18) = n^2 + 20n - (n^2 + 18n + 2n + 36) = n^2 + 20n - n^2 - 20n - 36 = -36 $$ - The difference is $-36$, so the absolute difference is always 36. 5. **(iii) Sum of the five numbers in the cross:** - The five numbers are: - Top left: $n$ - Top right: $n + 2$ - Center: $n + 10$ - Bottom left: $n + 18$ - Bottom right: $n + 20$ - Sum: $$ S = n + (n + 2) + (n + 10) + (n + 18) + (n + 20) = 5n + 50 $$ - To check if $S = 1715$ is possible: $$ 5n + 50 = 1715 \\ 5n = 1715 - 50 = 1665 \\ n = \frac{1665}{5} = 333 $$ - But $n$ must be a number in the grid from 1 to 37 (top left corner of the cross), so $n=333$ is impossible. **Therefore, the sum cannot be 1715.**