Table of integers
Is it possible to arrange infinite number of integers in the form of a table
(like a chessboard), so that every integer is the average of the 8 integers surrounding it?
Yes it is possible
a=b+c
b=d+e
c=f+g
d=h+i
e=j+k
f=l+m
g=n+o
Now suppose a is a middle number in 3 by 3 box
since any integer can be expressed as sum of two integers with infinite possibilities .following the above steps we can understand a can be expressed as sum of h,I,j,k,l,m,n,o .with each variable having infinite possibiles.now at any point we can find infinite integers which can surround a satisfying given condition.this can be used for any number in the table.we will find 8 integers which we have not used .now expanding table we will never be restricted to choose one number for the numbers surrounding the boundaries of table there will at least be a option of choosing 2 now since we have used limited numbers till now.there will be numbers having a required sum and uncommon with numbers we have used.so there will be no blockage in expanding our table
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