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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? 

 Jan 16, 2019
 #1
avatar+533 
0

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!

 Jan 16, 2019
 #2
avatar+57 
+1

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

TwentyFour  Jan 16, 2019
 #10
avatar+25 
+1

This answer is incorrect.

HelloWorld  Jan 16, 2019
 #3
avatar+533 
0

I GOT THE FIRST ONE?!!?!?!??!!??!!!

wow im so happy

also 1 3 6 10 is triangular numbers

 Jan 16, 2019
 #4
avatar+57 
+1

Yup! You should be able to do this pretty quickly now (:

TwentyFour  Jan 16, 2019
 #5
avatar+533 
0

wait what does the 1 3 6 10 mean is it the exponents or is it just n!*(n-1)!*(n-2)!*...?

 Jan 16, 2019
 #6
avatar+57 
+1

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.

TwentyFour  Jan 16, 2019
 #7
avatar+533 
0

ohhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh i see so if you were in 4th dimension, LOLOL it would be nc3!*... 

 Jan 16, 2019
 #8
avatar+57 
+1

Perfect for the introduction of the 3rd part of the problem.

 

What happens if you are in the mth dimension?

 

(:

TwentyFour  Jan 16, 2019
 #9
avatar+533 
0

wait but if you got so low, then in n choose m, n could be lower

 Jan 16, 2019

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