Puzzles/Analytical Puzzles/Clock Talk/Solution

Answers: 1: The hands pass 22 times in a 24-hour period.

Imagine we start at 00:01 and end at 24:01, the minute hand will cross the hour hand exacly 22 times. The first time a little after 01:05, second time - a little after 02:10, then after 03:15, 04:20, 05:25, 06:30, 07:35, 08:40, 09:45, the tenth time after 10:50 - almost at 10:55 and the 11th time after 11:55 in fact at 12:00. The whole sequence repeats for the next 12 hours.

2: Starting from 00:01, after 4 passes it will be a little after 04:20. More precisely 04:21:49.

After "H" hours and "M" minutes (where H<12 and M<60):

1. The hour hand has moved just a fraction of the first turn (in one 24-hour period it will make two 12-hour turns),

  it will be at ( H + M/60 )/12

2. The minute hand has made "H" complete turns and will be in its way to complete the next turn,

  it will be at ( M/60 )

3. To cross, the two hands ought to be at the same place:

  ( H + M/60 )/12 = ( M/60 )   =>   M = 11H/60

4. Therefore, before noon (remember, H<12) the two hands cross at:

  H=01, M=05.45 => 01:05:27 AM (because 0.45 minutes = 60(0.45) sec = 27 sec aprox.)
  H=02, M=10.91 => 02:10:54 AM
  H=03, M=16.36 => 03:16:21 AM
  H=04, M=21.82 => 04:21:49 AM
  H=05, M=27.27 => 05:27:16 AM
  H=06, M=32.73 => 06:32:43 AM
  H=07, M=38.18 => 07:38:10 AM
  H=08, M=43.64 => 08:43:38 AM
  H=09, M=49.09 => 09:49:05 AM
  H=10, M=54.55 => 10:54:32 AM
  H=11, M=60.00 => 12:00:00 PM (this is like 60 minutes after 11, which is actually noon)

5. After noon you can use the formula M = 11(H-12)/60 (H in a 24 hour format can be greater that 12) or assume that the cycle repeats itself (just interchange AM/PM).

Conclusion: they cross 22 times a day.

About the 4th pass, this is a little ambiguous. It depends whether you consider the first pass at 00:00:00 AM (H=0, M=0) or the last one at 12:00:00 AM (H=23, M=60).