Lattice hopping

Question

A bunny is hopping around the plane. She must

1. land on the following $25$ points on the main diagonal (multiples of 4): $(4, 4)$, $(8, 8)$, $(12, 12)$, $\ldots,$ $(100, 100)$, (The bunny can land on other points too, but the goal is to hit these multiples of four.)

2. only land on lattice points $\mathbb{Z}^2$,

3. only jump integer distances, and

4. move so no pair of jumps are parallel or perpendicular to each other. That means if the bunny jumps directly north for its first jump, it cannot jump directly north, south, east or west for any of the remaining jumps.

The bunny can start by jumping to any lattice point you wish. How can the bunny accomplish this task?

Answer 1

If $(a,b,c)$ is a Pythagorean triple, the bunny can do moves $(a,b)$ followed by $(-a,b)$ to move a displacement of $2b$ in any orthogonal direction.

Using triples of the form $(m^2-1,2m,m^2+1)$, we can obtain any even number $b=2m$, and so any orthogonal displacement $4m$. By moving $4(m+1)$ spaces, then $4m$ spaces the other way, we can move $4$ spaces.

With $m$ sufficiently large, we can guarantee that we haven't used $m$ and $m+1$ before, so this move isn't parallel or perpendicular to any previous move. (When $m$ is odd, the triple has a GCD of 2, so we have to care that its halved version hasn't been used either.)

The bunny can start at $(0,0)$ and repeatedly move $4$ steps and $4$ steps right to hit every required point.