Square Routes Re-Verticed

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 + i1, v + i2…, only allow a jump towards v + i1, v + i2… 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

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