IX OM - II - Task 2

Six equal disks are placed on a plane in such a way that their centers lie at the vertices of a regular hexagon with a side equal to the diameter of the disks. How many rotations will a seventh disk of the same size make while rolling externally on the same plane along the disks until it returns to its initial position?
Let circle $K$ with center $O$ and radius $r$ roll without slipping on a circle with center $S$ and radius $R$ (Fig. 16). The rolling without slipping means that different points of one circle successively coincide with different points of the other circle, and in this correspondence, the length of the arc between two points of one circle equals the length of the arc between the corresponding points of the other circle.
Consider two positions $K$ and $K_1$ of the moving circle, corresponding to centers $O$ and $O_1$ and points of tangency with the fixed circle $A$ and $B_1$; let $B$ be the point on the circumference of circle $K$ that, in the new position, passes to the point of tangency $B_1$, and let $\alpha$ and $\beta$ denote the radian measures of angles $ASB_1$ and $AOB$. Since the length of arc $AB_1$ of the fixed circle equals the length of arc $AB$ of the moving circle, we have $R \alpha = r \beta$. The radius $OA$ of the moving circle will take the new position $O_1A_1$, with $\measuredangle A_1O_1B_1 = \measuredangle AOB = \beta$. Draw the radius $O_1A_0$ parallel to $OA$; we get $\measuredangle A_0O_1B_1 = \measuredangle ASB_1 = \alpha$. The angle $A_0O_1A_1 = \alpha + \beta$ is equal to the angle through which circle $K$ has rotated when moving from position $K$ to position $K_1$. If $R = r$, then $\beta = \alpha$, and the angle of rotation is then $2 \alpha$.
Once the above is established, it is easy to answer the posed question.
The moving disk $K$ rolls successively on six given disks; the transition from one to another occurs at positions where disk $K$ is simultaneously tangent to two adjacent fixed disks. Let $K_1$ and $K_2$ be the positions where disk $K$ is tangent to fixed disk $N$ and to one of the adjacent disks (Fig. 17). Then $\measuredangle O_1SO_2 = 120^\circ$, so when moving from position $K_1$ to position $K_2$, the disk rotates by $240^\circ$. The disk will return to its initial position after $6$ such rotations, having rotated a total of $6 \cdot 240^\circ = 4 \cdot 360^\circ$, i.e., it will complete $4$ full rotations.
Note. We can more generally consider the rolling of a disk with radius $r$ on any curve $C$ (Fig. 18). When the point of tangency $A$ traverses an arc $AB_1$ of length $l$ on the curve, the angle through which the disk rotates is $\alpha + \beta$, where $\alpha$ is the angle between the lines $OA$ and $O_1B_1$, i.e., the angle between the normals to the curve $C$ at points $A$ and $B$, and $\beta = \frac{l}{r}$.