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Omni-MATH / Find all integers n for which each cell of n×n table can be filled with one of the letters I,M and O in such a way that: [LIST] [*] in each row and each column, one third of the en…

Problem

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problem

Find all integers nn for which each cell of n×nn \times n table can be filled with one of the letters I,MI,M and OO in such a way that: [LIST] [*] in each row and each column, one third of the entries are II, one third are MM and one third are OO; and?[/*] [*]in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are II, one third are MM and one third are OO.[/*] [/LIST] [b]Note.[/b] The rows and columns of an n×nn \times n table are each labelled 11 to nn in a natural order. Thus each cell corresponds to a pair of positive integer (i,j)(i,j) with 1≤i,j≤n1 \le i,j \le n. For n>1n>1, the table has 4n−24n-2 diagonals of two types. A diagonal of first type consists all cells (i,j)(i,j) for which i+ji+j is a constant, and the diagonal of this second type consists all cells (i,j)(i,j) for which i−ji-j is constant.
Plain-text mathematical notation (without MathML)
Find all integers n for which each cell of n×n table can be filled with one of the letters I,M and O in such a way that:
[LIST]
[*] in each row and each column, one third of the entries are I, one third are M and one third are O; and?[/*]
[*]in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are I, one third are M and one third are O.[/*]
[/LIST]
[b]Note.[/b] The rows and columns of an n×n table are each labelled 1 to n in a natural order. Thus each cell corresponds to a pair of positive integer (i,j) with 1≤i,j≤n. For n>1, the table has 4n−2 diagonals of two types. A diagonal of first type consists all cells (i,j)  for which i+j is a constant, and the diagonal of this second type consists all cells (i,j) for which i−j is constant.
Original LaTeX notation
Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that:
[LIST]
[*] in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and?[/*]
[*]in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.[/*]
[/LIST]
[b]Note.[/b] The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i,j)$ with $1 \le i,j \le n$. For $n>1$, the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i,j)$  for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i,j)$ for which $i-j$ is constant.

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Source and history

task_id
bb0a7556-afc0-58af-a590-e621e4e97ecc
task_key
test--bb0a7556-afc0-58af-a590-e621e4e97ecc
task_revision_id
1
config / split
/ test

Official source

initial import