# Help: Grid Walking

In a $3$x$3$ grid composed of $4$ vertical lines and $4$ horizontal lines; a beetle is moving from one intersecting point to another in one step. If the beetle moves from the bottom left corner to the top right corner within $8$ steps, and it never returned to the same intersecting point, what is the number of different paths the beetle can take?

Note by Ryan Merino
3 weeks, 4 days ago

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@Zakir Husain I got 36 for exactly 8 steps, I think I might have over counted, is there any mistake you see in my approach?

- 3 weeks, 1 day ago

There are total $36$ paths I found it using a Python program

- 3 weeks, 3 days ago

That I think would be for exactly 8 steps, including 6 steps will make it 36

- 3 weeks, 1 day ago

$Following\space are\space sequence\space of\space movements:$ $(0, 0)\to (1, 0)\to (2, 0)\to (3, 0)\to (3, 1)\to (3, 2)\to (2, 2)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (2, 0)\to (3, 0)\to (3, 1)\to (2, 1)\to (2, 2)\to (3, 2)\to (3, 3)$ $(0, 0)\to (1, 0)\to (2, 0)\to (3, 0)\to (3, 1)\to (2, 1)\to (2, 2)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (2, 0)\to (2, 1)\to (3, 1)\to (3, 2)\to (2, 2)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (2, 0)\to (2, 1)\to (2, 2)\to (1, 2)\to (1, 3)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (2, 0)\to (2, 1)\to (1, 1)\to (1, 2)\to (2, 2)\to (3, 2)\to (3, 3)$ $(0, 0)\to (1, 0)\to (2, 0)\to (2, 1)\to (1, 1)\to (1, 2)\to (2, 2)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (2, 0)\to (2, 1)\to (1, 1)\to (1, 2)\to (1, 3)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (1, 1)\to (2, 1)\to (3, 1)\to (3, 2)\to (2, 2)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (1, 1)\to (2, 1)\to (2, 2)\to (1, 2)\to (1, 3)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (1, 1)\to (2, 1)\to (2, 0)\to (3, 0)\to (3, 1)\to (3, 2)\to (3, 3)$ $(0, 0)\to (1, 0)\to (1, 1)\to (1, 2)\to (2, 2)\to (2, 1)\to (3, 1)\to (3, 2)\to (3, 3)$ $(0, 0)\to (1, 0)\to (1, 1)\to (1, 2)\to (1, 3)\to (2, 3)\to (2, 2)\to (3, 2)\to (3, 3)$ $(0, 0)\to (1, 0)\to (1, 1)\to (1, 2)\to (0, 2)\to (0, 3)\to (1, 3)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (1, 1)\to (0, 1)\to (0, 2)\to (1, 2)\to (2, 2)\to (3, 2)\to (3, 3)$ $(0, 0)\to (1, 0)\to (1, 1)\to (0, 1)\to (0, 2)\to (1, 2)\to (2, 2)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (1, 1)\to (0, 1)\to (0, 2)\to (1, 2)\to (1, 3)\to (2, 3)\to (3, 3)$ $(0, 0)\to (1, 0)\to (1, 1)\to (0, 1)\to (0, 2)\to (0, 3)\to (1, 3)\to (2, 3)\to (3, 3)$ $(0, 0)\to (0, 1)\to (1, 1)\to (2, 1)\to (3, 1)\to (3, 2)\to (2, 2)\to (2, 3)\to (3, 3)$ $(0, 0)\to (0, 1)\to (1, 1)\to (2, 1)\to (2, 2)\to (1, 2)\to (1, 3)\to (2, 3)\to (3, 3)$ $(0, 0)\to (0, 1)\to (1, 1)\to (2, 1)\to (2, 0)\to (3, 0)\to (3, 1)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (1, 1)\to (1, 2)\to (2, 2)\to (2, 1)\to (3, 1)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (1, 1)\to (1, 2)\to (1, 3)\to (2, 3)\to (2, 2)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (1, 1)\to (1, 2)\to (0, 2)\to (0, 3)\to (1, 3)\to (2, 3)\to (3, 3)$ $(0, 0)\to (0, 1)\to (1, 1)\to (1, 0)\to (2, 0)\to (3, 0)\to (3, 1)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (1, 1)\to (1, 0)\to (2, 0)\to (2, 1)\to (3, 1)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (1, 1)\to (1, 0)\to (2, 0)\to (2, 1)\to (2, 2)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (1, 1)\to (1, 0)\to (2, 0)\to (2, 1)\to (2, 2)\to (2, 3)\to (3, 3)$ $(0, 0)\to (0, 1)\to (0, 2)\to (1, 2)\to (2, 2)\to (2, 1)\to (3, 1)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (0, 2)\to (1, 2)\to (1, 3)\to (2, 3)\to (2, 2)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (0, 2)\to (1, 2)\to (1, 1)\to (2, 1)\to (3, 1)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (0, 2)\to (1, 2)\to (1, 1)\to (2, 1)\to (2, 2)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (0, 2)\to (1, 2)\to (1, 1)\to (2, 1)\to (2, 2)\to (2, 3)\to (3, 3)$ $(0, 0)\to (0, 1)\to (0, 2)\to (0, 3)\to (1, 3)\to (2, 3)\to (2, 2)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (0, 2)\to (0, 3)\to (1, 3)\to (1, 2)\to (2, 2)\to (3, 2)\to (3, 3)$ $(0, 0)\to (0, 1)\to (0, 2)\to (0, 3)\to (1, 3)\to (1, 2)\to (2, 2)\to (2, 3)\to (3, 3)$

- 3 weeks, 1 day ago

$(0,0)→(0,1)→(0,0)→(1,0)→(1,1)→(1,2)→(1,3)→(2,3)→(3,3)$

This path isn’t allowed because we come back to origin again, I think so without these ones there should be 36 paths with length 8

- 3 weeks, 1 day ago

OIC found new mistake ;)

- 3 weeks, 1 day ago

Edited!

- 3 weeks, 1 day ago

I don’t see any path like this

$(0,0) ➝ (0,1) ➝ (1,1) ➝ …$

One example of a complete path not present

$(0,0) ➝ (0,1) ➝ (1,1) ➝ (1,2) ➝ (2,2) ➝ (2,1) ➝ (3,1) ➝ (3,2) ➝ (3,3)$

(I do silly mistakes all the time, I might have not understood the question too)

- 3 weeks, 1 day ago

There was a problem in my program, I fixed it and found $56$ paths

- 3 weeks, 1 day ago

@Jeff Giff Now I edited My comment

- 3 weeks, 1 day ago