Problem 5. Koschei the Deathless gave Baba Yaga an electric broom. On the instrument panel of the electric broom, there is a display that shows the time in the format HH:MM (for example, 13:56) and the charge of the broom on a hundred-point scale (in whole numbers from 1 to 100). The broom consumes the charge evenly and is completely discharged in 10 hours. The display shows the maximum charge value as 100, and instead of showing 0, the broom lands due to lack of energy. At 00:00, Baba Yaga took off on a fully charged broom, and throughout the entire time until landing, the broom did not receive power. At what time during the flight do the charge value of the broom and the number of minutes on the display coincide?
Answer: $04: 52,05: 43,06: 35,07: 26,09: 09$.

Solution. In 10 hours (= 600 minutes), the charge is completely depleted, so if the charge value decreases, it will decrease by 1 again exactly 6 minutes later. For the first 4 hours, the charge value is $>60$, so during this time, the charge value and minutes definitely do not match. Next, we will create a table (see the figure), in which we will mark 60 minutes, divided into groups of six, and the charge values. The row that starts with a charge value of 60 corresponds to the fifth hour (time on the display 04:**), the row that starts with a charge value of 50 - the sixth hour, and so on. It remains to find the charge values that fall within the required time intervals and restore the time. Note that on the ninth hour, there will be no matches: the display will change from $[8: 1718]$ to [8:18 17], and the minutes will not match the charge value.

![](https://cdn.mathpix.com/cropped/2024_05_06_3414435e0bcdd156a13bg-3.jpg?height=831&width=1422&top_left_y=544&top_left_x=131)