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FoolForMath Group Title

Just another super easy problem, A \( 5\times 5 \) square is made of square tiles of dimensions \( 1\times 1 \). A mouse can leap along the diagonal or along the side of square tiles. In how many ways can the mouse reach the right lower corner vertex of the square from the lower left corner vertex of the square leaping exactly \(5\) times?

  • 2 years ago
  • 2 years ago

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  1. apoorvk Group Title
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    Is it 7 by any chance?

    • 2 years ago
  2. FoolForMath Group Title
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    No, but it's a multiple of 7 ;)

    • 2 years ago
  3. apoorvk Group Title
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    make that 14.. or am i lolling myself again?

    • 2 years ago
  4. Ishaan94 Group Title
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    idk why but i feel it's 21.

    • 2 years ago
  5. Ishaan94 Group Title
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    I havent solved it yet, just a guess.

    • 2 years ago
  6. apoorvk Group Title
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    |dw:1338132939113:dw| need one more.

    • 2 years ago
  7. Ishaan94 Group Title
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    Why did the tiled square got deleted? It was helpful.

    • 2 years ago
  8. Ishaan94 Group Title
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    Now I will have to draw on my notebook :/

    • 2 years ago
  9. experimentX Group Title
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    |dw:1338133230250:dw| Not so better than before!!

    • 2 years ago
  10. apoorvk Group Title
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    |dw:1338133213060:dw| okay 5 more - i learnt not to 'derive' answers lol.

    • 2 years ago
  11. apoorvk Group Title
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    BUGG!!!!^^

    • 2 years ago
  12. apoorvk Group Title
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    |dw:1338133822413:dw|

    • 2 years ago
  13. Ishaan94 Group Title
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    Instead of 5, we can try three and four to get a generalized pattern. Counting isn't the right way.

    • 2 years ago
  14. Ishaan94 Group Title
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    For me at least. I keep messing up my count.

    • 2 years ago
  15. apoorvk Group Title
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    |dw:1338134551987:dw|

    • 2 years ago
  16. TuringTest Group Title
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    |dw:1338134999071:dw|i made a picture :)

    • 2 years ago
  17. apoorvk Group Title
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    Cool^^

    • 2 years ago
  18. TuringTest Group Title
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    lol

    • 2 years ago
  19. TuringTest Group Title
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    ...now somebody analyze it

    • 2 years ago
  20. apoorvk Group Title
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    I got 21 - and am pretty sure.

    • 2 years ago
  21. TuringTest Group Title
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    I'd like to show it though. I wanna figure a way to count based on the number of horizontal moves the mouse makes, like there is only 1 possible path with 5 moves horizontal|dw:1338135897949:dw|4 not possible... how many ways can he do it if he goes 3 horizontal steps?\ at least that's how I'm thinking...

    • 2 years ago
  22. TuringTest Group Title
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    |dw:1338136237586:dw|I see 3 possibilities along the bottom and one if he goes along the middle totaling 4 now it would be nice to find a pattern rather than count for 2

    • 2 years ago
  23. TuringTest Group Title
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    no, there are more... I made a mistake

    • 2 years ago
  24. apoorvk Group Title
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    That is what I drew above - the black lines are the movements along the grid.

    • 2 years ago
  25. TuringTest Group Title
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    I think you are right @apoorvk I just can't prove it

    • 2 years ago
  26. Ishaan94 Group Title
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    How do we know it's 21?

    • 2 years ago
  27. Ishaan94 Group Title
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    Are you sure of your counting?

    • 2 years ago
  28. Ishaan94 Group Title
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    |dw:1338137956244:dw|

    • 2 years ago
  29. Ishaan94 Group Title
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    The triangle doesnt trace this path.

    • 2 years ago
  30. TuringTest Group Title
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    what path is that o-0 ? it's all corner moves, right?

    • 2 years ago
  31. apoorvk Group Title
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    It's 21 for this - I am sure of this - checked that. How do we generalise this though?

    • 2 years ago
  32. Ishaan94 Group Title
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    What do you mean by corner move?

    • 2 years ago
  33. TuringTest Group Title
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    " In how many ways can the mouse reach the right lower corner vertex from the lower left corner vertex of the square" he can't start in the middle of a square, only a corner, so he cannot move vertically at all

    • 2 years ago
  34. TuringTest Group Title
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    ...only diagonally

    • 2 years ago
  35. Ishaan94 Group Title
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    " A mouse can leap along the diagonal or along the side of square tiles."

    • 2 years ago
  36. TuringTest Group Title
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    yeah, but try it if he makes a vertical move he will never reach the lower left corner

    • 2 years ago
  37. TuringTest Group Title
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    he's got to stay on the grid is the point I think

    • 2 years ago
  38. Ishaan94 Group Title
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    |dw:1338138315764:dw|

    • 2 years ago
  39. TuringTest Group Title
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    |dw:1338138438753:dw|" ...from the lower left corner vertex of the square " I took that to mean that he starts at this point|dw:1338138495546:dw|

    • 2 years ago
  40. TuringTest Group Title
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    and how is this|dw:1338138529062:dw|moving "along the edge of the tile"? I'd say that's moving through the middle of it

    • 2 years ago
  41. Ishaan94 Group Title
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    oh

    • 2 years ago
  42. Ishaan94 Group Title
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    @FoolForMath is offline :/

    • 2 years ago
  43. TuringTest Group Title
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    he gave me a medal for my drawing, so I think that would mean I interpreted it correctly how did you read it @apoorvk ?

    • 2 years ago
  44. apoorvk Group Title
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    yeah ofcourse on the grid!! The instructions are pretty clear about that - it's about "vertex-to-vertex" jump, not from "spot-to-spot".

    • 2 years ago
  45. Ishaan94 Group Title
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    I see. I was doing it wrong all along :/ :( such a wasted effort :/

    • 2 years ago
  46. TuringTest Group Title
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    but that's great, now you can do it correctly for us @Ishaan94 :D

    • 2 years ago
  47. FoolForMath Group Title
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    21 is the right answer. The general solution is also amazing :D Thanks to M.SE I found this one: http://en.wikipedia.org/wiki/Motzkin_number http://math.stackexchange.com/questions/150420/

    • 2 years ago
  48. experimentX Group Title
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    lol ... that wasn't easy!! enlightening though!!

    • 2 years ago
  49. FoolForMath Group Title
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    I agree :D

    • 2 years ago
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