To determine the expected number of squares that a line segment is next to, we need to calculate the probability of each possible neighboring square and multiply it by the number of squares that have that probability.
Let's analyze the grid starting from an arbitrary position and calculate the number of neighboring squares for each segment in the grid.
In the figure below, we represent the number of neighboring squares for each segment using different colors:
1 (red)
2 (green)
3 (blue)
2 (green)
1 (red)
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| | | |
| R | G | G |
| | | |
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| | | |
| R | B | B |
| | | |
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| | | |
| R | G | G |
| | | |
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| | | |
| R | G | G |
| | | |
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| | | |
| R | G | G |
| | | |
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Based on the grid structure, we can deduce the following:
- Segments on the corners have 2 neighboring squares.
- Segments on the edges (excluding corners) have 3 neighboring squares.
- Segments in the interior have 4 neighboring squares.
Counting the number of each type of segment, we find:
- There are 4 corner segments.
- There are 12 edge segments.
- There are 9 interior segments.
Calculating the expected number of neighboring squares:
- Corner segments contribute 2 * 4 = 8 neighboring squares.
- Edge segments contribute 3 * 12 = 36 neighboring squares.
- Interior segments contribute 4 * 9 = 36 neighboring squares.
Adding up the contributions, we get a total of 8 + 36 + 36 = 80 neighboring squares.
Since there are a total of 30 segments in the grid (4 corners + 12 edges + 9 interior segments), the expected number of squares that a line segment is next to is 80 / 30 ≈ 2.67.
Therefore, the expected number of squares that a line segment is next to is approximately 2.67.
The grid below is made up of line segments, like the line segment in red. There are a number of paths that go from A to B in the grid, where every step goes to the right or up. If we choose a line segment at random, then what is the expected number of squares that it is next to?
1 answer