1. A necklace consists of 30 blue and a certain number of red beads. It is known that on both sides of each blue bead there are beads of different colors, and one bead away from each red bead there are also beads of different colors. How many red beads can be in this necklace? (The beads in the necklace are arranged cyclically, that is, the last one is adjacent to the first.)
Answer: 60.

Solution. It is obvious that blue beads appear in the necklace in pairs, separated by at least one red bead. Let there be $n$ red beads between two nearest pairs of blue beads. We will prove that $n=4$. Clearly, $n \leqslant 4$, since the middle one of five consecutive red beads does not satisfy the condition of the problem. For $n<4$, there are three possible situations:

CCKKKCC, CCKKCC, CCKCC.

In the first case, the middle red bead does not satisfy the condition, and in the other cases, all red beads do not satisfy the condition.

Thus, pairs of blue beads must be separated by four red beads. It is clear that such a necklace satisfies the condition of the problem. In it, the number of red beads is twice the number of blue beads, that is, there are 60.