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