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Complementary Counting Challenge:

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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
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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
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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
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HelloWorld  Jan 16, 2019
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I GOT THE FIRST ONE?!!?!?!??!!??!!!

wow im so happy

also 1 3 6 10 is triangular numbers

Jan 16, 2019
#4
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Yup! You should be able to do this pretty quickly now (:

TwentyFour  Jan 16, 2019
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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
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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
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ohhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh i see so if you were in 4th dimension, LOLOL it would be nc3!*...

Jan 16, 2019
#8
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Perfect for the introduction of the 3rd part of the problem.

What happens if you are in the mth dimension?

(:

TwentyFour  Jan 16, 2019
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wait but if you got so low, then in n choose m, n could be lower

Jan 16, 2019