a) On a digital clock, hours are displayed using four digits. For example, when showing 00:00 we know it is midnight, and when showing 23:59 we know it is one minute before midnight. How many times a day do all four digits shown are even?

b) A clock, with hour, minute, and second hands, makes a "plim" every time a hand passes another on the dial. The three hands only overlap at noon or midnight. What is the number of plims recorded in a certain day, from 12 hours and 1 second to 23 hours, 59 minutes, and 59 seconds?
Solution

a) Let's say the display is made up of 4 digits. The first digit can only be occupied by the numbers 0, 1, or 2. The second digit depends on the first: when the first is 0 or 1, any digit can appear, and when the first is 2, 0, 1, 2, or 3 can appear. The third digit only allows the digits from 0 to 5. Additionally, the fourth digit allows the occurrence of all digits. In the first digit, there are two cases of possible numbers:

i) Suppose the first digit is occupied by 0; in the second, 0, 2, 4, 6, or 8 can appear; in the third, 0, 2, or 4; and in the last, 0, 2, 4, 6, or 8. Therefore, there are $1 \times 5 \times 3 \times 5 = 75$ possible times.

ii) Suppose the first digit is occupied by 2; in the second digit, only 0 or 2 are possible; in the third digit, 0, 2, or 4 can appear; and in the fourth, 0, 2, 4, 6, or 8. Therefore, there are $1 \times 2 \times 3 \times 5 = 30$ moments with the rule applied.

In the end, we have $75 + 30 = 105$ times per day.
b) There are three situations:

i) The hour hand is overtaken by the second hand every minute, resulting in 60 times per hour, or in total, the hour hand is overtaken by the second hand $12 \times 60 = 720$ times, excluding the time $00:00$, which results in 719 plinks.

ii) The minute hand is overtaken by the second hand every minute from 1 to 59, occurring 59 times, and in the given period, this situation occurs $12 \times 59 = 708$ times.

iii) The hour hand is overtaken by the minute hand once per hour, i.e., it will occur 12 times, excluding the case of $0 \text{h}$, it occurs 11 times.

Therefore, in total, there will be $719 + 708 + 11 = 1438$ plinks.