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FoolForMath

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?

  • one year ago
  • one year ago

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  1. apoorvk
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    Is it 7 by any chance?

    • one year ago
  2. FoolForMath
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    No, but it's a multiple of 7 ;)

    • one year ago
  3. apoorvk
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    make that 14.. or am i lolling myself again?

    • one year ago
  4. Ishaan94
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    idk why but i feel it's 21.

    • one year ago
  5. Ishaan94
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    I havent solved it yet, just a guess.

    • one year ago
  6. apoorvk
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    |dw:1338132939113:dw| need one more.

    • one year ago
  7. Ishaan94
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    Why did the tiled square got deleted? It was helpful.

    • one year ago
  8. Ishaan94
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    Now I will have to draw on my notebook :/

    • one year ago
  9. experimentX
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    |dw:1338133230250:dw| Not so better than before!!

    • one year ago
  10. apoorvk
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    |dw:1338133213060:dw| okay 5 more - i learnt not to 'derive' answers lol.

    • one year ago
  11. apoorvk
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    BUGG!!!!^^

    • one year ago
  12. apoorvk
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    |dw:1338133822413:dw|

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

    • one year ago
  14. Ishaan94
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    For me at least. I keep messing up my count.

    • one year ago
  15. apoorvk
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    |dw:1338134551987:dw|

    • one year ago
  16. TuringTest
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    |dw:1338134999071:dw|i made a picture :)

    • one year ago
  17. apoorvk
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    Cool^^

    • one year ago
  18. TuringTest
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    lol

    • one year ago
  19. TuringTest
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    ...now somebody analyze it

    • one year ago
  20. apoorvk
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    I got 21 - and am pretty sure.

    • one year ago
  21. TuringTest
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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...

    • one year ago
  22. TuringTest
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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

    • one year ago
  23. TuringTest
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    no, there are more... I made a mistake

    • one year ago
  24. apoorvk
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    That is what I drew above - the black lines are the movements along the grid.

    • one year ago
  25. TuringTest
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    I think you are right @apoorvk I just can't prove it

    • one year ago
  26. Ishaan94
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    How do we know it's 21?

    • one year ago
  27. Ishaan94
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    Are you sure of your counting?

    • one year ago
  28. Ishaan94
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    |dw:1338137956244:dw|

    • one year ago
  29. Ishaan94
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    The triangle doesnt trace this path.

    • one year ago
  30. TuringTest
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    what path is that o-0 ? it's all corner moves, right?

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

    • one year ago
  32. Ishaan94
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    What do you mean by corner move?

    • one year ago
  33. TuringTest
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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

    • one year ago
  34. TuringTest
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    ...only diagonally

    • one year ago
  35. Ishaan94
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    " A mouse can leap along the diagonal or along the side of square tiles."

    • one year ago
  36. TuringTest
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    yeah, but try it if he makes a vertical move he will never reach the lower left corner

    • one year ago
  37. TuringTest
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    he's got to stay on the grid is the point I think

    • one year ago
  38. Ishaan94
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    |dw:1338138315764:dw|

    • one year ago
  39. TuringTest
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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|

    • one year ago
  40. TuringTest
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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

    • one year ago
  41. Ishaan94
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    oh

    • one year ago
  42. Ishaan94
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    @FoolForMath is offline :/

    • one year ago
  43. TuringTest
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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 ?

    • one year ago
  44. apoorvk
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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".

    • one year ago
  45. Ishaan94
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    I see. I was doing it wrong all along :/ :( such a wasted effort :/

    • one year ago
  46. TuringTest
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    but that's great, now you can do it correctly for us @Ishaan94 :D

    • one year ago
  47. FoolForMath
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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/

    • one year ago
  48. experimentX
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    lol ... that wasn't easy!! enlightening though!!

    • one year ago
  49. FoolForMath
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    I agree :D

    • one year ago
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