Oddly Spaced Numbers
Can we a write the numbers 1 to 9 in a line from left to right so
that between each number \(i\) and
\(i+1\) (i.e.., between 1 and 2,
between 2 and 3, \(\ldots\), and
between 8 and 9) there is an odd number of numbers? If yes, how? If not
why?
“The number of numbers between \(a\)
and \(b\)" includes neither \(a\) nor \(b\). For example, in \(\langle 1,3,5,2,4,8,9,6,7 \rangle\) there
is one number between 4 and 5, and there are two numbers between 1 and
2.
Link to the problem on Twitter: https://twitter.com/Riazi_Cafe/status/1684084582600298496
The answer is negative.
If such an ordering existed, then for every pair \(i\) and \(i+1\), the parity of the positions of \(i\) and \(j\) in the permutation would be the same.
Thus either all numbers would be placed on odd positions or all numbers
would be placed on even positions; Neither of which is possible.