A nice Maths problem?

You have a rectangle that is m x n units. When you draw a diagonal from one corner to the other, you get to identical right-angled triangles. The length of the diagonal, which is the number of squares it passes through (h) is given by Pythagoras' Theorem:

h^2 = o^2 + a^2, giving

h^2 = m^2 + n^2

implying that h = square root of (m^2 + n^2)

Still thinking about it, but formula would need to allow answer to drop at points, even when m and n increase. For example 2x3 the diagonal goes through 4 squares, yet 3x3 only goes through 3.

STILL struggling. You only need a formula for prime numbers for the larger dimension, say m, as all other sizes of m are multiples of these. If n was the larger dimension you could simply swap m & n round so a formula for m alone would apply.

Say the line runs from (x,y)= (0,0) to (m,n), so it satisfies nx = my. At a lattice point (k1,k2), n*k1 = m*k2 but since m,n are relatively prime k1 is a multiple of m and k2 a multiple of n.

In other words except for the start and end points, the line crosses only lines but no lattice points. That means a new square is entered exactly when the line crosses a (vertical or horizontal) line so the total number of squares passed through is just:

= 1 + Number of lines crossed

= 1 + Horizontal lines crossed + Vertical lines crossed

= 1 + (m-1) + (n-1)

= m + n - 1

Your m square units and n square units are confusing.

Do you mean,

|_m|n_| ?