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

To prove Victor's statement, let's label the nine consecutive numbers arranged in a 3-by-3 spiral format. We can denote these numbers as: 1, 2, 3, 4, 5, 6, 7, 8, 9.

Arranging them in a 3-by-3 grid:

4  5  6
3  0  7
2  1  8

In this arrangement, let's analyze the totals of the first and second columns:

• First Column Total: 4 + 3 + 2 = 9
• Second Column Total: 5 + 0 + 1 = 6

Thus, the first column total (9) is indeed one less than the second column total (6).

Let's generalize this result: Assume we have nine consecutive numbers starting from any integer n:

• The numbers will be: n, n+1, n+2, n+3, n+4, n+5, n+6, n+7, n+8

Arranging them in a 3-by-3 grid:

(n+3)  (n+4)  (n+5)
(n+2)  (n)   (n+6)
(n+1)  (n+0)  (n+7)

Calculating the first and second columns:

• First Column Total: (n+3) + (n+2) + (n+1) = 3n + 6
• Second Column Total: (n+4) + n + 0 = 2n + 4

We can see that:

First Column Total = 3n + 6 Second Column Total = 2n + 4

The difference is:

Total of First Column - Total of Second Column = (3n + 6) - (2n + 4) = n + 2

Since we are considering consecutive numbers,

This formula shows that for any nine consecutive numbers, the total in the first column will always be one less than that in the second column, thus proving Victor's statement.