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- An algorithm generates square patterns like this: For any value n, there is a grid of nxn alternating dark-and-light squares, so like-colored squares never touch except for their vertices. These are placed on a background whose color is identical to the light-colored squares, so a dark square alone is surrounded on all sides by a light color. So:
- n = 1: n = 2: n = 3: n = 4: n = 5:
- X X X X X X X X X
- X X X X X X
- X X X X X X X
- X X X X
- X X X
- and so on. Find all groups within these shapes, where a group is defined as a square pattern where all 4 sides are bordered by opposite colors. This function will be called f(x). So:
- f(1) = 1 dark = 1 total group
- Darks:
- +-+
- |X|
- +-+
- f(2) = 2 dark = 2 total groups
- Darks:
- +-+
- |X|
- +-+-+
- |X|
- +-+
- f(3) = 5 dark = 5 total groups
- Darks:
- +-+ +-+
- |X| |X|
- +-+-+-+
- |X|
- +-+-+-+
- |X| |X|
- +-+ +-+
- f(4) = 8 dark + 2 light + 1 2x2 subgroup = 11 total groups
- Darks: Lights: 2x2s:
- +-+ +-+
- |X| |X| X X X X
- +-+-+-+-+ +-+ +---+
- |X| |X| X| |X |X |X
- +-+-+-+-+ +-+-+ | |
- |X| |X| X| |X X| X|
- +-+-+-+-+ +-+ +---+
- |X| |X| X X X X
- +-+ +-+
- f(5) = 13 dark + 4 light + 4 2x2 subgroups + 1 3x3 subgroup = 22 total groups
- Darks: Lights: 2x2s: 3x3s:
- +-+ +-+ +-+
- |X| |X| |X| X X X X X X X X X
- +-+-+-+-+-+ +-+ +-+-+-+ +-----+
- |X| |X| X| |X |X| |X| |X X|
- +-+-+-+-+-+ +-+-+-+ +-+-+-+ | |
- |X| |X| |X| X| |X| |X X| |X| |X X| X |X
- +-+-+-+-+-+ +-+-+-+ +-+-+-+ | |
- |X| |X| X| |X |X| |X| |X X|
- +-+-+-+-+-+ +-+ +-+-+-+ +-----+
- |X| |X| |X| X X X X X X X X X
- +-+ +-+ +-+
- And so on.
- THE CHALLENGE IS TO FIND A FUNCTION f THAT WILL FIND THE CORRECT OUTPUT (number of total groups) FOR ANY COUNTING NUMBER x.
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