Track / Stage 4 / 180 of 340 #440 of 1964

Problem 440

AMC 12 late, AIME early

Combinatorics Difficulty 4.9 Find the answer

Consider $C$ .

$N$ digits - ones and twos - are arranged in a circle. An "image" is a number formed by several consecutive digits (either clockwise or counterclockwise). For what smallest value of $N$ can all four-digit numbers, whose representation contains only the digits 1 and 2, be among the images?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Note that the images of the numbers 1111, 2112, and 2122 cannot have common units, and the images of the numbers 2222, 1221, and 1211 cannot have common twos. Therefore, if all these numbers are among the images, then there must be at least 14 digits around the circle - 7 ones and 7 twos. The equality $N=14$ is possible, and the corresponding example of the arrangement of digits is shown in the figure.

## Answer

14.00

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.