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Reversing Minute Hand Clock (Posted on 2023-09-19) Difficulty: 3 of 5
An analog clock's hour hand moves normally, but its minute hand reverses direction whenever it meets the hour hand. Initially, the clock reads 12:00 and both hands are moving clockwise.
Where will the minute hand be when the hour hand has completed a 12 hour circuit, and which direction will it be going?

No Solution Yet Submitted by Larry    
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Solution solution Comment 1 of 1
When the minute hand is going prograde it takes the two hands 12/11 of an hour from meeting to meeting. When it's going retrograde, it takes 12/13 of an hour.

A pair of these takes 288/143 of an hour. In 12 hours that's 143/24 =~ 5.95833333333333 times this pairing, so it should be in the second portion (retrograde portion) of the pair of motions.

That's (6 - 143/24) of the 288/143 hours for the full cycle or 12/143 of an hour, which is 12/143 / (12/13) of the retrograde portion of the cycle or 1/11 of that retrograde period. 1/11 of the backward motion, taken of is 360/11 degrees clockwise from the 12 o'clock position or just over 32.7 degrees, or a little past the 1 o'clock position, headed toward the 1 and 12 o'clock positions, as it's retrograding.

  Posted by Charlie on 2023-09-19 08:27:49
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