How Many Distinct Islands Can Exist in an nxn Grid with Multiple Colors?

[TenaliRaman]

1>Given a nxn grid , we colour the grid with 2 colours such that we get the maximum number of distinct islands. What is the maximum number of distinct islands possible for a given nxn grid?
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2>For 3-colours?
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3>Is generalisation possible to m-colours??
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[clarification]
given a 3x3 grid and 2 colours then this colouration *
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1||2||1
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2||2||2
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1||2||1
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gives us 5 islands (the 4 1's and the 1 group of 2's)
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We consider horizontal adjacency,vertical adjacency and also diagonal adjacency when grouping to get islands.
[/clarification]
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[TenaliRaman]

has anyone had any progress on this??

All i could judge abt this question was that it was remotely equivalent to the kings problem so the answer must be very much similar. Though a rigorous proof would take the meat!

-- AI

 

[M98Ranger]

1) The maximum number of distinct islands possible for a given nxn grid with 2 colors is n^2/2 + 1. This is because, in a 2-color grid, each cell can either be colored with one color or the other, giving us a total of 2^n possible combinations. However, since we are looking for distinct islands, we cannot count the entire grid as one island, so we subtract 1 from the total number of combinations. Therefore, the maximum number of distinct islands is n^2/2 + 1.
2) For 3 colors, the maximum number of distinct islands possible for a given nxn grid would be n^2/3 + 1. This is because each cell can now be colored with one of the three colors, giving us a total of 3^n combinations. Again, we subtract 1 to account for the entire grid being counted as one island.
3) The generalization to m-colors is possible, and the maximum number of distinct islands for a given nxn grid would be n^2/m + 1. This is because with m colors, we have m^n possible combinations, but we still need to subtract 1 to account for the entire grid being counted as one island.