The 64 squares of an 8×8 chessboard are filled with positive integers in such a way that each
integer is the average of the integers on the neighbouring squares. (Two squares are neighbours
if they share a common edge or a common vertex. Thus a square can have 8, 5 or 3 neighbours
depending on its position). Show that all the 64 integer entries are in fact equal.

Question

The 64 squares of an 8×8 chessboard are filled with positive integers in such a way that each
integer is the average of the integers on the neighbouring squares. (Two squares are neighbours
if they share a common edge or a common vertex. Thus a square can have 8, 5 or 3 neighbours
depending on its position). Show that all the 64 integer entries are in fact equal.

goformit100 · Answer

@TeamJakeRosati_06 @mikaela19900630

anonymous · Answer

http://tech.groups.yahoo.com/group/mathforfun/message/1064 try this

anonymous · Answer

Hmmm, I would pick the smallest number of the grid , it's neighbored numbers must be the same size or larger because I have picked the smallest.If it  is larger than the picked number the average will be larger than the picked number. Therefore all the neighbors must be equal to the picked number.

goformit100 · Answer

Thank you Sir and Madam

anonymous · Answer

your very welcome