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# Beautiful intricate math problem

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Hello! Here is a really fun problem I came up with.

You are given a perfect square $$n.$$

Define a superpair in $$n$$ as a $$2(n-1)$$-tuple, $$(x_1, x_2, x_3, ... x_{2(n-1)})$$such that $$x_1\cdot x_2 \cdots x_{n-1} = x_n \cdot x_{n+1} \cdots x_{2(n-1)},$$ with $$1 \leq x_i \leq n$$.  In other words, all elements must be between $$1$$ and $$n$$ and the product of its first $$n-1$$ elements equals the product of its last $$n-1$$ elements.)

Find the set $$S$$ of values of $$n$$ that guarantee that it is possible to form list of $$n$$ superpairs in $$n$$ such that every number that appears in your superpair list appears exactly twice in your superpair list.

For example, we know that $$n=3$$ is in $$S$$ because we can satisfy the conditions with the following superpair list:

$$(1,6,2,3)$$

$$(2,4,1,8)$$

$$(3,8,6,4)$$

Here, $$1\cdot 6 = 2 \cdot 3,$$ $$2 \cdot 4 = 1 \cdot 8,$$ and $$3 \cdot 8 = 6 \cdot 4.$$ Additionally, every number present in the the superpair list, ($$1,2,3,4,6,8$$) appears exactly twice.

Notice that not all numbers from $$1$$ to $$n$$ need to be in your superpair list. Rather, you must only ensure that the ones that are in your superpair list, appear exactly twice. Hint: try using exactly $$n^2-n$$ distinct numbers in your superpair list.

I hope you enjoy!

Jan 22, 2022

#1
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What is your solution to this problem? I'm curious.

Jan 23, 2022
#2
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"What is your solution to this problem? I'm curious."

"Hint: try using exactly  n^2 -n distinct numbers in your superpair list.

How can you give us a hint if you do not know what the answer is ?

If you came up with it yourself and you do not know what the answer is then how can you claim it is 'beautiful'?

Melody  Jan 23, 2022