## 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

1. apoorvk Group Title

Is it 7 by any chance?

2. FoolForMath Group Title

No, but it's a multiple of 7 ;)

3. apoorvk Group Title

make that 14.. or am i lolling myself again?

4. Ishaan94 Group Title

idk why but i feel it's 21.

5. Ishaan94 Group Title

I havent solved it yet, just a guess.

6. apoorvk Group Title

|dw:1338132939113:dw| need one more.

7. Ishaan94 Group Title

Why did the tiled square got deleted? It was helpful.

8. Ishaan94 Group Title

Now I will have to draw on my notebook :/

9. experimentX Group Title

|dw:1338133230250:dw| Not so better than before!!

10. apoorvk Group Title

|dw:1338133213060:dw| okay 5 more - i learnt not to 'derive' answers lol.

11. apoorvk Group Title

BUGG!!!!^^

12. apoorvk Group Title

|dw:1338133822413:dw|

13. Ishaan94 Group Title

Instead of 5, we can try three and four to get a generalized pattern. Counting isn't the right way.

14. Ishaan94 Group Title

For me at least. I keep messing up my count.

15. apoorvk Group Title

|dw:1338134551987:dw|

16. TuringTest Group Title

17. apoorvk Group Title

Cool^^

18. TuringTest Group Title

lol

19. TuringTest Group Title

...now somebody analyze it

20. apoorvk Group Title

I got 21 - and am pretty sure.

21. TuringTest Group Title

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 Group Title

|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 Group Title

no, there are more... I made a mistake

24. apoorvk Group Title

That is what I drew above - the black lines are the movements along the grid.

25. TuringTest Group Title

I think you are right @apoorvk I just can't prove it

26. Ishaan94 Group Title

How do we know it's 21?

27. Ishaan94 Group Title

Are you sure of your counting?

28. Ishaan94 Group Title

|dw:1338137956244:dw|

29. Ishaan94 Group Title

The triangle doesnt trace this path.

30. TuringTest Group Title

what path is that o-0 ? it's all corner moves, right?

31. apoorvk Group Title

It's 21 for this - I am sure of this - checked that. How do we generalise this though?

32. Ishaan94 Group Title

What do you mean by corner move?

33. TuringTest Group Title

" 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 Group Title

...only diagonally

35. Ishaan94 Group Title

" A mouse can leap along the diagonal or along the side of square tiles."

36. TuringTest Group Title

yeah, but try it if he makes a vertical move he will never reach the lower left corner

37. TuringTest Group Title

he's got to stay on the grid is the point I think

38. Ishaan94 Group Title

|dw:1338138315764:dw|

39. TuringTest Group Title

|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 Group Title

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 Group Title

oh

42. Ishaan94 Group Title

@FoolForMath is offline :/

43. TuringTest Group Title

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 Group Title

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 Group Title

I see. I was doing it wrong all along :/ :( such a wasted effort :/

46. TuringTest Group Title

but that's great, now you can do it correctly for us @Ishaan94 :D

47. FoolForMath Group Title

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 Group Title

lol ... that wasn't easy!! enlightening though!!

49. FoolForMath Group Title

I agree :D