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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?
No, but it's a multiple of 7 ;)
make that 14.. or am i lolling myself again?
idk why but i feel it's 21.
I havent solved it yet, just a guess.
|dw:1338132939113:dw| need one more.
Why did the tiled square got deleted? It was helpful.
Now I will have to draw on my notebook :/
|dw:1338133230250:dw| Not so better than before!!
|dw:1338133213060:dw| okay 5 more - i learnt not to 'derive' answers lol.
Instead of 5, we can try three and four to get a generalized pattern. Counting isn't the right way.
For me at least. I keep messing up my count.
|dw:1338134999071:dw|i made a picture :)
...now somebody analyze it
I got 21 - and am pretty sure.
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...
|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
no, there are more... I made a mistake
That is what I drew above - the black lines are the movements along the grid.
I think you are right @apoorvk I just can't prove it
How do we know it's 21?
Are you sure of your counting?
The triangle doesnt trace this path.
what path is that o-0 ? it's all corner moves, right?
It's 21 for this - I am sure of this - checked that. How do we generalise this though?
What do you mean by corner move?
" 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
" A mouse can leap along the diagonal or along the side of square tiles."
yeah, but try it if he makes a vertical move he will never reach the lower left corner
he's got to stay on the grid is the point I think
|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|
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
@FoolForMath is offline :/
he gave me a medal for my drawing, so I think that would mean I interpreted it correctly how did you read it @apoorvk ?
yeah ofcourse on the grid!! The instructions are pretty clear about that - it's about "vertex-to-vertex" jump, not from "spot-to-spot".
I see. I was doing it wrong all along :/ :( such a wasted effort :/
but that's great, now you can do it correctly for us @Ishaan94 :D
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/
lol ... that wasn't easy!! enlightening though!!