Start with a point in the middle of a square. Allow it to make a series of, say, eight jumps towards the vertices of the square, but with one restriction: it can’t jump towards the same vertex twice in a row. When the point has made the eight jumps, mark its position. If you do this for every possible route, the result will look like this:

Ban jump towards same vertex

And here’s a different restriction: the point can’t jump towards the vertex immediately to the left of the vertex it has just jumped towards:

Ban jump towards *v* + 1

And here it can’t jump towards the vertex diagonally opposite the vertex it has just jumped towards:

Ban jump towards *v* + 2

Now allow the point to jump not just towards the vertices, but towards points midway between the vertices. And expand and reverse the restrictions: instead of not allowing a jump towards *v* + i_{1}, *v* + i_{2}…, only allow a jump towards *v* + i_{1}, *v* + i_{2}… Some interesting shapes appear:

Jump must be towards *v*, *v* + 1 or *v* + 2 (one point between vertices)

*v*, *v* + 1 or *v* + 6

*v*, *v* + 2 or *v* + 3

*v*, *v* + 2 or *v* + 4

*v*, *v* + 2 or *v* + 6

*v*, *v* + 3 or *v* + 4

*v*, *v* + 3 or *v* + 5

*v*, *v* + 2 or *v* + 7

*v* + 1, *v* + 4 or *v* + 7

*v*, *v* + 1 or *v* + 6 (two points between vertices)

*v*, *v* + 2 or *v* + 4

*v*, *v* + 2 or *v* + 6

*v*, *v* + 2 or *v* + 9

*v*, *v* + 3 or *v* + 6

*v*, *v* + 3 or *v* + 8

*v*, *v* + 4 or *v* + 8

*v*, *v* + 5 or *v* + 7

*v* , *v* + 6 or *v* + 11

*v* + 1, *v* + 5 or *v* + 6

*v* + 1, *v* + 2 or *v* + 10

*v* + 1, *v* + 6 or *v* + 10

*v* + 1, *v* + 6 or *v* + 11

*v* + 2, *v* + 6 or *v* + 10

Elsewhere other-posted:

• Square Routes

• Square Routes Revisited

• Square Routes Re-Revisited

• Square Routes Re-Re-Revisited