Victor says If any nine consecutive numbers are arranged in ascending order in this spiral on a 3-by-3 grid, the total of the first column will always be one less than the total of the second column - OCR - GCSE Maths - Question 12 - 2019 - Paper 1

The 3-by-3 grid, when filled in a spiral order, appears as follows:

 n       n+1     n+2
 n+7     n+8     n+3
 n+6     n+5     n+4

In this arrangement, the first column consists of the numbers $n$, $n+7$, and $n+6$, while the second column has $n+1$, $n+8$, and $n+5$.

Now, calculating the sums:

• Total of the first column:

$T_1 = n + (n+7) + (n+6) = 3n + 13$

• Total of the second column:

$T_2 = (n+1) + (n+8) + (n+5) = 3n + 14$

To see if Victor's statement holds, compare the two totals:

$T_2 - T_1 = (3n + 14) - (3n + 13) = 1$

Thus, it confirms that the total of the first column is always one less than the total of the second column.

Therefore,

$T_1 = T_2 - 1$

This proves that Victor's assertion is correct for any arrangement of nine consecutive numbers in a 3-by-3 spiral grid.