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Omni-MATH / Let n>1 be a given integer. An n×n×n cube is composed of n³ unit cubes. Each unit cube is painted with one colour. For each n×n×1 box consisting of n² unit cubes (in any of the thr…

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problem

Let n>1n > 1 be a given integer. An n×n×nn \times n \times n cube is composed of n3n^3 unit cubes. Each unit cube is painted with one colour. For each n×n×1n \times n \times 1 box consisting of n2n^2 unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get 3n3n sets of colours, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of nn, the maximal possible number of colours that are present.
Plain-text mathematical notation (without MathML)
Let n>1 be a given integer. An n×n×n cube is composed of n³ unit cubes. Each unit cube is painted with one colour. For each n×n×1 box consisting of n² unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get 3n sets of colours, split into three groups according to the orientation.

It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of n, the maximal possible number of colours that are present.
Original LaTeX notation
Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get $3n$ sets of colours, split into three groups according to the orientation.

It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colours that are present.

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Source and history

task_id
345e7331-2dc1-5b17-bc91-1138bd103298
task_key
test--345e7331-2dc1-5b17-bc91-1138bd103298
task_revision_id
1
config / split
/ test

Official source

initial import