Distinct Islands in the Matrix

  • Thread starter TenaliRaman
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In summary, the conversation discusses the maximum number of distinct islands that can be obtained by coloring a nxn grid with 2 colors, as well as the possibility of generalizing this to grids with more colors. The example of a 3x3 grid with 2 colors is given, where 5 islands are obtained by considering horizontal, vertical, and diagonal adjacency for grouping. One person mentions the similarity to the king's problem and suggests a rigorous proof is needed.
  • #1
TenaliRaman
644
1
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 *
*
1||2||1
--------
2||2||2
--------
1||2||1
*
gives us 5 islands (the 4 1's and the 1 group of 2's)
*
We consider horizontal adjacency,vertical adjacency and also diagonal adjacency when grouping to get islands.
[/clarification]
*
 
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  • #2
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
 
  • #3

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.
 

1. What are distinct islands in the matrix?

Distinct islands in the matrix refer to unique, isolated regions within a larger matrix that have different characteristics or properties compared to the surrounding areas. These islands can vary in size and can be identified based on various factors such as color, shape, or composition.

2. How do distinct islands form in a matrix?

Distinct islands can form in a matrix through a variety of processes such as erosion, sedimentation, or volcanic activity. These processes can create variations in the matrix, resulting in the formation of isolated regions with distinct characteristics.

3. What is the significance of studying distinct islands in the matrix?

Studying distinct islands in the matrix can provide valuable insights into the geological history and processes that have shaped the matrix. It can also help in identifying potential natural resources or understanding the impact of human activities on the environment.

4. Can distinct islands in the matrix move or change over time?

Yes, distinct islands in the matrix can move or change over time due to various factors such as weathering, tectonic movements, or human activities. These changes can alter the characteristics of the islands and their surrounding areas.

5. How do scientists identify and study distinct islands in the matrix?

Scientists use various methods such as satellite imagery, aerial photography, and field surveys to identify and study distinct islands in the matrix. They also analyze data from geological maps, topographic maps, and remote sensing techniques to understand the characteristics and changes of these islands.

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