no_three_in_line_5x5.md•1.92 kB
# No-Three-in-Line Problem: Verification for n=5
## Problem Statement
Consider a 5×5 grid of points with coordinates (i,j) where i,j ∈ {0,1,2,3,4}. This gives us 25 possible positions arranged in a square grid pattern.
**Task:** Verify that it is possible to place exactly 10 points on this 5×5 grid such that no three of the selected points are collinear (lie on the same straight line).
## Constraints
- Points must be placed at integer grid coordinates only
- "No three in line" means no three selected points lie on the same straight line, regardless of the line's slope
- This includes horizontal lines, vertical lines, diagonal lines, and lines with any other slope (like 1/2, 2/1, 1/3, 1/4, 3/2, etc.)
## What to Verify
1. **Existence:** Find a configuration of 10 points on the 5×5 grid that satisfies the no-three-in-line constraint
2. **Completeness:** Check all possible lines that could pass through three or more points in your configuration to confirm none contain three selected points
3. **Optimality:** Verify that 10 is indeed the maximum number of points that can be placed (optional)
## Example Grid Layout
```
(0,4) (1,4) (2,4) (3,4) (4,4)
(0,3) (1,3) (2,3) (3,3) (4,3)
(0,2) (1,2) (2,2) (3,2) (4,2)
(0,1) (1,1) (2,1) (3,1) (4,1)
(0,0) (1,0) (2,0) (3,0) (4,0)
```
## Lines to Consider
The 5×5 grid contains 32 distinct lines with 3 or more collinear points:
- 5 horizontal lines
- 5 vertical lines
- 2 main diagonal lines
- 20 other diagonal lines with various slopes
Some examples of lines that must be avoided:
- **Horizontal:** [(0,0), (1,0), (2,0), (3,0), (4,0)]
- **Vertical:** [(0,0), (0,1), (0,2), (0,3), (0,4)]
- **Diagonal:** [(0,0), (1,1), (2,2), (3,3), (4,4)]
- **Slope 1/2:** [(0,0), (2,1), (4,2)]
- **Slope 2:** [(0,0), (1,2), (2,4)]
**Question:** Can you find such a configuration of 10 points and prove that no three of them are collinear?