Volchenkov S.G.

$N$ digits - ones and twos - are arranged in a circle. An image is called a number formed by several consecutive digits (clockwise or counterclockwise). For what smallest value of $N$ can all four-digit numbers, whose notation contains only the digits 1 and 2, be among the images?
![](https://cdn.mathpix.com/cropped/2024_05_06_1aa9346639cd54e59f2ag-10.jpg?height=183&width=180&top_left_y=452&top_left_x=939)

Notice that the images of the numbers 1111, 2112, and 2122 cannot have common units, while the images of the numbers 2222, 1221, and 1211 cannot have common twos. Therefore, if all these numbers appear 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