5. The magical clock, in addition to the usual pair of hands, has a second pair that is symmetrical to the first at every moment relative to the vertical axis. It is impossible to determine which hands are real from a photograph of the clock. Furthermore, just like with ordinary clocks, it is impossible to distinguish morning from evening using the magical clock. Therefore, the same photograph of the clock can correspond to several different times (for example, 1:15, $10:45$, and $22:45$ look the same as shown on the right).

![](https://cdn.mathpix.com/cropped/2024_05_06_d4b15693c622237d278eg-3.jpg?height=134&width=134&top_left_y=253&top_left_x=1783)

A robot takes several photographs of the clock over the course of a day (from 0:00 to 24:00). It remembers the order in which the photographs were taken but not the time they were taken. Sometimes, from such a series of photographs, it is possible to determine the exact time some of them were taken; such photographs will be called definite. If, however, there are several moments when a photograph could have been taken (even considering the other photographs in the series), it is called indefinite. For example, in the series of photographs shown on the right, photograph №2 is definite (it was taken at 9:00), while photograph №4 is indefinite (it could have been taken at either $16:00$ or 20:00).
![](https://cdn.mathpix.com/cropped/2024_05_06_d4b15693c622237d278eg-3.jpg?height=154&width=528&top_left_y=688&top_left_x=1388)

Suppose there is a series of 100 photographs taken over the course of a day, none of which look the same, and none of which were taken at 0:00, $6:00$, $12:00$, $18:00$, or 24:00. What is the minimum number of indefinite photographs that can be among them?

(A. A. Tsel)


Solution. See the solution to problem 6 for 10th grade.

Instead of the last paragraph of the solution to problem 6 for 10th grade, it is sufficient to provide an example with exactly three undefined photographs and prove that there are exactly three. For example, let the first three photos be taken at 5:50, 11:40, 17:30, and the rest in the interval from 20:00 to 23:00. Then, from the appearance of the photographs, it can be determined that the first one was taken no earlier than 5:50, which means the second one was taken no earlier than 11:40, and the third one no earlier than 17:30, and finally, the rest after 18:00, where only one option remains for them, meaning they are defined.

## Answer: 3.

Criteria. 1 point for providing an example with exactly 3 undefined photographs, +1 additional point if it is proven that the example is correct.

1 point for stating the proposition "each photograph corresponds to four possible times: $t, 12-t, 12+t, 24-t$."