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Crossed Ends

Age 11 to 14 Challenge Level:

Mohit, Harry, Jimbob, Sanjay,Imo and Jazzy all found that

When you add one pair of opposite numbers it is always equal to the other pair.
i.e. If you add N+S, W+E you get the same answer, and that answer is twice the number in the middle of the cross, whatever the size of the cross and the grid.
If you add N+S+E+W you always get 4x middle number.

Kang from The Garden International School sent us a clear solution to the problem:

Adding opposite pairs on a cross:
Always, the result will be the same with each pair.

Using algebra, it will be easy to explain and understand.
In a cross, the pairs are (up + down) & (left + right).
If the middle square is x, right will be x+1 and left will be x-1.
On a size 10 grid, up is x-10 and down is x+10.
They all add up to 2x.

The explanation for a bigger cross is the same, but with bigger numbers of + and - (assuming always the opposite sides of the cross are of equal length).

Adding adjacent pairs on a cross:
It is the same, but there is a difference and it is always the same.

If it is added together such as (N+E) & (S+W),
N = x-10
E = x+1
S = x+10
W = x-1
(x-10) + (x+1) = 2x-9
(x+10) + (x-1) = 2x+9
The difference between the adjacent pairs = (2x+9) - (2x-9) = 18
For other pair of adjacent sides, it works in the same way.

Harry from Brewood Middle School explained it differently:

To find a solution we need a key:
X=number at the top of the cross
G=Grid size
H=Height of cross (from top to middle)
W=Width of cross (from end to middle)
L=Left number
R=Right number
B=Bottom number
The Formula for finding B (if you only know X) is: 2GH+X.
Therefore the formula for adding X+B = 2GH+2X = (X+HG-W)+(X+HG+W) = L+R
Therefore, no matter where the cross is L+R = X+B

Similar reasoning was given by Andy and Lily Li from Garden International School. Well done to you all.