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How many arrangements of the numbers \(1, 2, 3, \dots, 7\) are there where the sum of any two adjacent numbers is odd?

 

Can someone start me off? I'll try to do the rest :)

 Feb 25, 2021
 #1
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There are a total of 5040 arrangements. 

 

Keep these relationships in mind:

 

even + odd = odd

odd + even = odd

odd + odd = even

even + even = odd

 

To have an odd sum of two adjacent numbers, they must have opposite parity (one even, one odd).

 

If I'm entirely honest, this looks like an interesting problem to solve. Even I haven't figured it out yet! I hope this gets you started.

 Feb 25, 2021

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