6. 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_e219a7e95d389188eca9g-2.jpg?height=132&width=131&top_left_y=962&top_left_x=1782)

A robot takes several photographs of the clock over the course of one 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 No. 2 is definite (it was taken at 9:00), while photograph No. 4 is indefinite (it could have been taken at either 16:00 or 20:00).
![](https://cdn.mathpix.com/cropped/2024_05_06_e219a7e95d389188eca9g-2.jpg?height=160&width=512&top_left_y=1415&top_left_x=1388)

Suppose there is a series of 100 photographs taken over the course of one 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. How many indefinite photographs can there be among them?

(A. A. Tsel)


Answer: 3 or 100.

Solution. First, note that for each photograph, the set of possible times has the form $\{t ; 12-t ; 12+t ; 24-t\}(0<t<6)$.

Consider the following photographs: $A$ - the one taken closest to 6:00; $B$ - the one taken closest to 12:00; $C$ - the one taken closest to 18:00.

If all these photographs are different, then each of them can be "symmetrically reflected" relative to the corresponding time (for example, if the first photograph was taken at 5:50, it could also have been taken at 6:10, which does not affect the correctness of the sequence of times). Thus, we have at least three uncertain photographs.

If any two of them coincide, then there exists a 12-hour interval in which only one photograph was taken (that very one that coincides): for example, if it was taken at the moment $6+t_{1}=12-t_{2}$, then there are no other photographs in the interval from $6-t_{1}$ to $12+t_{2}$. Then all photographs can fit into the period from 0:00 to 12:00, or from 12:00 to 24:00, meaning all photographs are uncertain.

These variants are schematically shown in the graphs. On the horizontal axis - the actual time of creation of the photographs; on the vertical axis - the minimum time at which the clock looks as in the photo; the red marks show the shots $A, B, C$, and arrows indicate their "uncertainty".
![](https://cdn.mathpix.com/cropped/2024_05_06_e219a7e95d389188eca9g-3.jpg?height=288&width=1714&top_left_y=96&top_left_x=202)

We will prove that in the case when not all photographs are uncertain, there are exactly three uncertain photographs. Indeed, the three selected photographs $(A, B, C)$ are different. Then for each of them, there are only two possible times: if for any of them (for example, A) there are at least three possible moments, then the interval between the extreme such moments is at least 12 hours, which allows all frames to fit into a 12-hour period (which means all shots are uncertain). Note that all photographs taken before shot $A$ were taken before 6:00 (since the other permissible times for them are later than both permissible times for photo $A$). Similarly, we establish that the photos taken between shots $A$ and $B$ were taken between 6 and 12 hours (otherwise, they are either taken earlier than both possibilities for $A$, or later than both possibilities for $B$), photos between $B$ and $C$ - between 12 and 18 hours, and finally, photos taken after $C$ - after 18 hours. Thus, all other shots turned out to be certain.

Criteria. One point for proving the possibilities of 3 and 100. Two points for proving that less than 3 is not possible.