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FoolForMath

  • 2 years ago

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?

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  1. apoorvk
    • 2 years ago
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    Is it 7 by any chance?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  17. apoorvk
    • 2 years ago
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    Cool^^

  18. TuringTest
    • 2 years ago
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    lol

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

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

  21. TuringTest
    • 2 years ago
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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...

  22. TuringTest
    • 2 years ago
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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

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

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

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

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

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

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

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

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

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

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

  33. TuringTest
    • 2 years ago
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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

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

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

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

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

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

  39. TuringTest
    • 2 years ago
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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|

  40. TuringTest
    • 2 years ago
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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

  41. Ishaan94
    • 2 years ago
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    oh

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

  43. TuringTest
    • 2 years ago
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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 ?

  44. apoorvk
    • 2 years ago
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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".

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

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

  47. FoolForMath
    • 2 years ago
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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/

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

  49. FoolForMath
    • 2 years ago
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    I agree :D

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