+62 There is an n by n square with n^2 unit squares. How many ways can we put numbers from 1 to n^2 on the grid such that the numbers from left to right and top to bottom go from least to greatest?
What if the grid is a n by n by n Cube?
+532 lets take a smaller case. 1x1 would only have 1 case.
now for 2x2. 1 and 4 must go in the corners, and 2 and 3 can be alternated 2 ways.
now for 3x3. 1 and 9 go in the corners, and then 8 and 7 border the 9, and 2 and 3 border the 1. now you can arrange the 4, 5, and 6 any way you wish, which makes 3! * 2 * 2 ways, or 24.
for 4x4, apply the same logic, and then you will have a diagonal, and two small diagonals. 4, 5, and 6 go in the top diagonal, 13, 12, 11 go in the bottom one, and the rest go in the middle, for 4!*3!*3!*2!*2!, or 3456.
you can now see that it is 1! for 1x1, then 2! for 2x2, then 3!*2!*2! for 3x3, and then 4!*3!*3!*2!*2!. so, if you had a nxn square, then n+1xn+1 square would be (n+1)!*(n!)*(n!)*(n-1)!*....3!*2!*2!*1!*1!, so for nxn square, it would be n!*((n-1)!)^2*((n-2)!)^2*...(3!)^2*(2!)^2*(1!)^2.
i may be wrong, i probably am, but somebody PLEASE check my work.
as for n by n by n, i am comepletely confuzzled O.O
HOPE THIS HELPED!
+62 First Answer is correct (:
Second answer is just like the first, except you are not adding 1, but it goes from
1, to 3 to 6, to 10....
That should give a pretty good hint (:
-24
+532 I GOT THE FIRST ONE?!!?!?!??!!??!!!
wow im so happy
also 1 3 6 10 is triangular numbers
+532 wait what does the 1 3 6 10 mean is it the exponents or is it just n!*(n-1)!*(n-2)!*...?
+62 We still have 1!*2!*3! and so forth, but this time we have triangular numbers, so it would start off like 1!*3!*6!*10!........ and so forth.
+532 ohhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh i see so if you were in 4th dimension, LOLOL it would be nc3!*...
+62 Perfect for the introduction of the 3rd part of the problem.
What happens if you are in the mth dimension?
(:
