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Crossed Ends

Age 11 to 14 Challenge Level:

These crosses can be drawn on number grids of various sizes.

Add opposite pairs of orange numbers (i.e. north + south, east + west).
Notice anything? Try a few more.

Now try adding opposite pairs on crosses with just two lines of symmetry, like these:

Experiment with grids of various sizes. What do you notice? Can you explain your findings?
A sheet of number grids is available here

What happens if you add the orange squares in adjacent pairs? (try N + W, S + E )
Can you predict in advance how the totals will relate to each other?
What does it depend on?
Is it the same if you added them the other way round? (i.e. N + E, S + W)

Can you explain your findings?