Word Problem (The Digital Clock and the Mirror) Help, PLEASE?

Question

As I look in the bathroom mirror while I'm shaving, I can see the reflection of the digital clock in my bedroom. Sometimes the time I see in the mirror (a reflection of the actual time) is the same as the actual time. How many times does this happen per day? (Ignore the colon because I can't see that anyway--the clock is too far away.)

Please show/explain each step!

Thanks!

Answer 1

0 flips to 0
1 flips to 1
2 flips to 5
5 flips to 2
8 flips to 8

If the hour is single digit (between 1 and 9), then:
-The middle digit must be its own reversal. Only 3 numbers have this property, namely 0, 1, and 8. However, 8 cannot be the middle digit, as there are only 60 minutes in an hour. So we have 2 valid possibilities for the middle digit.
-Also, the last digit must be the reversal of the first. Only 4 numbers between 1 and 9 have this property (1, 2, 5, 8).
-Thus, we count 2*4 = 8.
(1:01, 1:11, 2:05, 2:15, 5:02, 5:12, 8:08, 8:18)

If the hour is 10, then the minutes must be 01 (10:01)

If the hour is 11, then the minutes must be 11 (11:11)

If the hour is 12, then the minutes must be 51 (12:51)

8+3 = 11

Answer 2

I'm not sure that you can calculate this. I think you have to count it.

1:01
1:11
1:21
1:31
1:41
1:51

2:02
2:12

Hmm. I've changed my mind. It happens 6 times per hour for the hours from 1-9, or 54 times.

10:01
11:11
12:21

And once each hour for the remaining three hours, for a total of 57 times each 12 hour period.

If you use a standard clock, where you have 4:00 am and 4:00 pm, then it happens 114 times per day.

If you use military time, it happens again at 13:31, 14:41, and 15:51, for a total of 60 times per day.

Answer 3

♠ I suppose that your clock shows like [ab:ba]; now exclude 2,3,4,5,6,7,9 - they are non-symmetrical digits; thus 0,1,8 are left; exclude a=8 and b=8, too, as a<3, b<6;
♣ b=0, then a= 0,1; or k1=2!;
b=1, then a=1,0; or k2=2!; k=k1+k2=4;