Beelines

Stage: 4 Challenge Level: Challenge Level:1

This problem is about straight lines joining the origin to different points (with integer coordinates) in the first quadrant. Here are some examples:
Grid

For some of these points one coordinate is a factor of the other, for some the coordinates have a common factor, but for others the coordinates are coprime (that is they have no common factor except 1).

The lines cross the grid in different ways:

  • some lines form neat diagonals across the squares of the grid (for example the line from the origin to (5,5)),
  • others create diagonals of rectangles (for example the line from the origin to (2,6) creates a diagonal which crosses two identical rectangles,
  • some (for example the line from the origin to (11,1)) forms a single diagonal.

Can you find any relationships between the number of squares that lines cross and the coordinates of their end points and explain why they work?

Can you find any relationships between the number of grid lines that lines cross and the coordinates of their end points and explain why they work?

Can you describe any relationships between the coordinates of the end points of lines, the lengths of the lines and the lengths of the diagonals of any rectangles they cross?
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Published October 2000,January 2009.