My friend sent me the original question. Then he asked if 1-12 can be arranged in a way so that any three consecutive numbers have a sum that is not larger than 20. We guess the answer is no, since we tried many times and could not find a solution. However, we wonder why it is. Any help would be appreciated!
1 Answer
The arrangement is indeed impossible.
As the answer to the question you linked mentions, the sums of each segment of three adjacent numbers all themselves have to add up to $3 (1 + 2 + \cdots + 12) = 234$. So the mean of the sums-of-three is 19.5. If no individual three-number segment has a sum above 20, then at least six of the segments must have sums of 20. Moreover, having two consecutive equal sums-of-three is impossible: if one part of the wheel has the numbers $a\,b\,c\,d$ in order where $a + b + c = b + c + d$, then $a = d$, but the wheel can't repeat numbers. So sums-of-three have to alternate between 20 and 19.
Now suppose the wheel has seven adjacent numbers $a\,b\,c\,d\,e\,f\,g$ where $a + b + c = c + d + e = e + f + g = 20$ and $b + c + d = d + e + f = 19$. Then $d = a - 1$ (by subtracting $b + c + d = 19$ from $a + b + c = 20$) and likewise $d = g - 1$ (by subtracting $d + e + f = 19$ from $e + f + g = 20$). So $a = g$; that is, we have a repeated number, a contradiction.
(As a final note, this method generalizes to show that any wheel with numbers $1$ through $2n$, $n \geq 4$, must have three adjacent numbers that add up to at least $3n + 3$.)