The nine entries of a 3×3 grid are filled with −1, 0, or 1. Prove that...

The nine entries of a 3×3 grid are filled with −1, 0, or 1. Prove that among the eight resulting sums (three columns, three rows, or two diagonals) there will always be two that add to the same number.

Solutions

Expert Solution

There are possible sums and they are . These are obtained when a "place" (column,row or diagonal) contains the entries

ii) (or any permutation of it)

iii) (or any permutation of it)

iv)

v) (or any permutation of it)

vi) (or any permutation of it)

vii)

There are "places" to be filled ( columns, rows and diagonals). Thus by pigeonhole principle, it follows that at least two of the places would contain the same sum.

Colby Messinger answered 1 year ago

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