Help: Grid Walking

In a 33x33 grid composed of 44 vertical lines and 44 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 88 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
4 months, 1 week 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?

Jason Gomez - 4 months, 1 week ago

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There are total 3636 paths I found it using a Python program

Zakir Husain - 4 months, 1 week ago

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

Jason Gomez - 4 months, 1 week ago

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Following are sequence of movements:Following\space are\space sequence\space of\space movements: (0,0)(1,0)(2,0)(3,0)(3,1)(3,2)(2,2)(2,3)(3,3)(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)(1,0)(2,0)(3,0)(3,1)(2,1)(2,2)(3,2)(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)(1,0)(2,0)(3,0)(3,1)(2,1)(2,2)(2,3)(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)(1,0)(2,0)(2,1)(3,1)(3,2)(2,2)(2,3)(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)(1,0)(2,0)(2,1)(2,2)(1,2)(1,3)(2,3)(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)(1,0)(2,0)(2,1)(1,1)(1,2)(2,2)(3,2)(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)(1,0)(2,0)(2,1)(1,1)(1,2)(2,2)(2,3)(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)(1,0)(2,0)(2,1)(1,1)(1,2)(1,3)(2,3)(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)(1,0)(1,1)(2,1)(3,1)(3,2)(2,2)(2,3)(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)(1,0)(1,1)(2,1)(2,2)(1,2)(1,3)(2,3)(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)(1,0)(1,1)(2,1)(2,0)(3,0)(3,1)(3,2)(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)(1,0)(1,1)(1,2)(2,2)(2,1)(3,1)(3,2)(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)(1,0)(1,1)(1,2)(1,3)(2,3)(2,2)(3,2)(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)(1,0)(1,1)(1,2)(0,2)(0,3)(1,3)(2,3)(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)(1,0)(1,1)(0,1)(0,2)(1,2)(2,2)(3,2)(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)(1,0)(1,1)(0,1)(0,2)(1,2)(2,2)(2,3)(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)(1,0)(1,1)(0,1)(0,2)(1,2)(1,3)(2,3)(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)(1,0)(1,1)(0,1)(0,2)(0,3)(1,3)(2,3)(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)(0,1)(1,1)(2,1)(3,1)(3,2)(2,2)(2,3)(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)(0,1)(1,1)(2,1)(2,2)(1,2)(1,3)(2,3)(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)(0,1)(1,1)(2,1)(2,0)(3,0)(3,1)(3,2)(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)(0,1)(1,1)(1,2)(2,2)(2,1)(3,1)(3,2)(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)(0,1)(1,1)(1,2)(1,3)(2,3)(2,2)(3,2)(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)(0,1)(1,1)(1,2)(0,2)(0,3)(1,3)(2,3)(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)(0,1)(1,1)(1,0)(2,0)(3,0)(3,1)(3,2)(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)(0,1)(1,1)(1,0)(2,0)(2,1)(3,1)(3,2)(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)(0,1)(1,1)(1,0)(2,0)(2,1)(2,2)(3,2)(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)(0,1)(1,1)(1,0)(2,0)(2,1)(2,2)(2,3)(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)(0,1)(0,2)(1,2)(2,2)(2,1)(3,1)(3,2)(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)(0,1)(0,2)(1,2)(1,3)(2,3)(2,2)(3,2)(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)(0,1)(0,2)(1,2)(1,1)(2,1)(3,1)(3,2)(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)(0,1)(0,2)(1,2)(1,1)(2,1)(2,2)(3,2)(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)(0,1)(0,2)(1,2)(1,1)(2,1)(2,2)(2,3)(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)(0,1)(0,2)(0,3)(1,3)(2,3)(2,2)(3,2)(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)(0,1)(0,2)(0,3)(1,3)(1,2)(2,2)(3,2)(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)(0,1)(0,2)(0,3)(1,3)(1,2)(2,2)(2,3)(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 - 4 months, 1 week ago

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(0,0)(0,1)(0,0)(1,0)(1,1)(1,2)(1,3)(2,3)(3,3)(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 - 4 months, 1 week ago

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OIC found new mistake ;)

Zakir Husain - 4 months, 1 week ago

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Edited!

Zakir Husain - 4 months, 1 week ago

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I don’t see any path like this

(0,0)(0,1)(1,1) (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) (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 - 4 months, 1 week ago

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There was a problem in my program, I fixed it and found 5656 paths

Zakir Husain - 4 months, 1 week ago

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@Jeff Giff Now I edited My comment

Zakir Husain - 4 months, 1 week ago

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