417. The lengths of the edges of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ are 1 cm. Find the smallest distance between the points of the circles, one of which is inscribed in the base of the cube $A B C D$, and the other passes through the vertices $A, C$, and $B_{1}$.

20th All-Union Olympiad,
1986 (Ulyanovsk)

| Class | 1st day | 2nd day |  |
| :---: | :---: | :---: | :---: |
| 8 | 41841942042 | 430431 | 432 |
| 9 | 42242342442 | 434 433b | 435 |
| 10 | $426 \quad 427 \quad 428 \quad 42$ | 437438 | 439 |
417. Answer: $(\sqrt{3}-\sqrt{2}) / 2$. The considered circles lie on two concentric spheres: one circumscribed around the cube and the other touching all its edges. The smallest distance is equal to the difference in the radii of these spheres.