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?

#Combinatorics

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
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Comments

@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?

Jason Gomez - 2 months, 3 weeks ago

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

Zakir Husain - 2 months, 3 weeks ago

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

Jason Gomez - 2 months, 3 weeks ago

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

Zakir Husain - 2 months, 3 weeks 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

Jason Gomez - 2 months, 3 weeks ago

OIC found new mistake ;)

Zakir Husain - 2 months, 3 weeks ago

Edited!

Zakir Husain - 2 months, 3 weeks 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)

Jason Gomez - 2 months, 3 weeks ago

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

Zakir Husain - 2 months, 3 weeks ago

@Jeff Giff Now I edited My comment

Zakir Husain - 2 months, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$