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'Adjacent Factors' printed from https://nrich.maths.org/
Answer: 9
Build up a long list
$6$
$2$ $6$ $3$
$4$ $2$ $6$ $3$ $9$
$8$ $4$ $2$ $6$ $3$ $9$ $1$
$8$ $4$ $2$ $6$ $3$ $9$ $1$ $5$ $10$
All numbers except $7$ used
Can we also use $7$?
$7$ should go at the end because it can only go next to $1$
$7$ $1$ $?$
$5$ will only fit if $5$ comes next (using the $1$) or if $5$ comes last, after $10$, which would come after $2$
$5$ next:
$7$ $1$ $5$ $10$ $2$ $?$
If $8$ or $4$ used next, $3$, $6$, $9$ won't fit
If $6$ used next, $8$ and $4$ won't fit
$5$ last:
$7$ $1$ $...$ $?$ $2$ $10$ $5$
Exactly the same problem as before now happens before the $2$
We can't use the $7$. We can only use 9 numbers.
The diagram below shows all of the possible connections that can be used.
There are many other sequences of nine numbers that follow the rule. Can you find them all?