Mohit, Harry, Jimbob, Sanjay,Imo and Jazzy
all found that
When you add one pair of opposite numbers it is always equal to the
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
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
H=Height of cross (from top to middle)
W=Width of cross (from end to middle)
The Formula for finding B (if you only know X) is:
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.