In a 12-hour clock that runs correctly, how many times do the second, minute, and hour hands of the clock coincide, in a...

Question

In a 12-hour clock that runs correctly, how many times do the second, minute, and hour hands of the clock coincide, in a 12-hour duration from 3 PM in a day to 3 AM the next day?

A. 11
B. 12
C. 144
D. 2

Accepted Answer

In 12 hours, the minute and hour hands coincide 11 times; all three hands (including the second hand) coincide only once (at 12:00:00 midnight). The question was officially excluded (answer key: X). The correct computed value closest to the intent is 11. For all three clock hands to coincide, we need the second, minute, and hour hands at the same angular position simultaneously. The minute and hour hands coincide 11 times in any 12-hour period (at approximately every 65.45 minutes). For all three hands to coincide, this must happen at a moment when the second hand is also at the same position. Since the second hand completes a full revolution every 60 seconds, it will be at any given position for only an instant — but the minute-hour coincidences last only an instant too. The probability that all three coincide is effectively zero for practical clock times. In an ideal mathematical clock, all three hands coincide only when all are at 12:00:00 (noon or midnight), i.e., once per 12 hours. From 3 PM to 3 AM, they coincide at 12:00:00 AM midnight, which is 1 time. However, the conventional GATE answer for 'three-hand coincidence' equates it to the 11 times the minute and hour hands coincide, treating the second hand as always present. The question is marked 'X' (excluded) in the answer key.

In a 12-hour clock that runs correctly, how many times do the second, minute, and hour hands of the clock coincide, in a 12-hour duration from 3 PM in a day to 3 AM the next day? A. 11 B. 12 C. 144 D. 2