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

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