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# Crossed Ends

##### Age 11 to 14Challenge 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).

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.