Let $n \geq 3$ be an integer. On a circle, there are $n$ points. Each of these points is labeled with a real number that is at most 1. Each number is equal to the absolute value of the difference of the two numbers that immediately precede it in a clockwise direction.

Determine the maximum possible value, in dependence on $n$, that the sum of all numbers can take.

(Walther Janous)

Answer. For $n$ divisible by 3, the sum can maximally take the value $2n / 3$, for all other $n$ it is always 0.
First, we observe that all numbers are absolute values and as such must be greater than or equal to zero.

If we fix two adjacent numbers $x_{1}$ and $x_{2}$ on the circle, we can uniquely calculate all other numbers from them. We will now try to insert a few numbers and see what happens, see Table 1.

| $\mathbf{x}_{\mathbf{1}}$ | $\mathbf{0 , 8}$ | $\mathbf{0 , 1}$ | $\mathbf{0 , 8}$ | $\mathbf{0 , 4}$ | $\mathbf{0 , 8}$ | $\mathbf{0 , 6}$ | $\mathbf{0 , 3}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $\mathbf{x}_{\mathbf{2}}$ | $\mathbf{0 , 1}$ | $\mathbf{0 , 8}$ | $\mathbf{0 , 4}$ | $\mathbf{0 , 8}$ | $\mathbf{0 , 6}$ | $\mathbf{0 , 8}$ | $\mathbf{0}$ |
| $x_{3}$ | 0,7 | 0,7 | 0,4 | 0,4 | 0,2 | 0,2 | 0,3 |
| $x_{4}$ | 0,6 | 0,1 | 0 | 0,4 | 0,4 | 0,6 | 0,3 |
| $x_{5}$ | 0,1 | 0,6 | 0,4 | 0 | 0,2 | 0,4 | 0 |
| $x_{6}$ | 0,5 | 0,5 | 0,4 | 0,4 | 0,2 | 0,2 | 0,3 |
| $x_{7}$ | 0,4 | 0,1 | 0 | 0,4 | 0 | 0,2 | 0,3 |
| $x_{8}$ | 0,1 | 0,4 | 0,4 | 0 | 0,2 | 0 | 0 |
| $x_{9}$ | 0,3 | 0,3 | 0,4 | 0,4 | 0,2 | 0,2 | 0,3 |
| $x_{10}$ | 0,2 | 0,1 | 0 | 0,4 | 0 | 0,2 | 0,3 |
| $x_{11}$ | 0,1 | 0,2 | 0,4 | 0 | 0,2 | 0 | 0 |
| $x_{12}$ | 0,1 | 0,1 | 0,4 | 0,4 | 0,2 | 0,2 | 0,3 |
| $x_{13}$ | 0 | 0,1 | 0 | 0,4 | 0 | 0,2 | 0,3 |

Table 1: Problem 3, Experiments

It is noticeable: If we choose two different positive numbers as starting numbers, the numbers tend to get smaller and smaller. This is because the absolute value of the difference of two positive numbers is always smaller than the larger of the two numbers. In particular, the larger of the two starting numbers will never appear again: The third number is genuinely smaller than the larger of the two, and the fourth number is the same (because the third, as the difference of two different numbers, is positive, and we can repeat the same argument). From here, zero can also occur (as in the example with starting numbers 0.8 and 0.4), but the absolute value of the difference is at least smaller or equal to the larger of the two numbers whose difference was formed. Thus, all subsequent numbers are smaller than the larger starting number. However, since we have to go around the circle and end up back at the starting numbers, this is not allowed.

Therefore, the two starting numbers must either be identical or one of them must be 0, meaning we start with $x, x$ or with $x, 0$ or with $0, x$ (with a real number $x$). In all these cases, a pattern $x, x, 0$ repeats itself (as in the last column of the table).

For $x=0$, i.e., all zeros, this always works. If, however, $x>0$, $n$ must be divisible by 3 so that the pattern fits after one round around the circle.

In this case, the sum of all numbers is $x \cdot \frac{2}{3} \cdot n$, which is largest for $x=1$.

(Birgit Vera Schmidt)