100 rows, 100 columns

Number Theory Level 3

Write exactly one integer from 1 to 3 into each cell of a $100\times 100$ square. Is it possibe, that the sum of the 100 numbers in any row, column or diagonal is different?

Yes, it's possible. No, it's not possible.

1 solution

Marco Brezzi
Aug 22, 2017

Since every cell is at least $1$ and at most $3$ , the sum of every row/column/diagonal is at least $100$ and at most $300$ . So there are $201$ possible sums (all the natural numbers between $100$ and $300$ ).

But since there are $100$ rows, $100$ columns and $2$ diagonals we need at least $202$ different sums, so $\boxed{\text{it's not possible}}$

100 rows, 100 columns

1 solution

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