The Assembly Hall Clock That Gains Four Minutes Every Hour
St. Xavier's assembly hall clock runs fast. In every 60 minutes of real time it advances 64 minutes, so it gains exactly 4 minutes per real hour, steadily and without jolts. On Monday at 7:00 a.m. the caretaker sets it to the correct time. Nobody touches it again for the rest of the week. On Wednesday at 7:00 a.m. by her phone, a teacher finds the clock reading 10:12 a.m. and complains that 'a slightly fast clock must still show the right time twice a day anyway'. The caretaker, who is preparing for a reasoning olympiad, is not convinced. School timings are fixed against real time: assembly at 8:00 a.m., lunch bell at 12:30 p.m. and dismissal at 3:00 p.m. The clock has no second hand and no alarm; it simply runs continuously at the constant fast rate described above, and it never stops or resets.
1. What does the clock read at 7:00 p.m. on Monday, measured against real time? Show the conversion.
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From 7:00 a.m. to 7:00 p.m. is 12 real hours = 720 real minutes.
In every 60 real minutes the clock advances 64 minutes, so in 12 real hours it advances 12 × 64 = 768 minutes.
768 minutes = 12 hours 48 minutes.
Starting from a correct 7:00 a.m., the clock therefore reads 7:00 a.m. + 12 h 48 min = 7:48 p.m.
Equivalently, count only the gain: 12 real hours × 4 minutes = 48 minutes fast, so the display is 7:00 + 0:48 ahead of the truth, i.e. 7:48 p.m. Both routes agree, which is a useful self-check.
2. When will the clock next show the correct time? Give the day and the real time.
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The clock agrees with real time whenever its total gain is a whole number of 12-hour cycles, because a 12-hour display cannot distinguish 7:00 from 19:00.
Required gain = 12 hours = 720 minutes.
Gain per real hour = 4 minutes, so the elapsed real time needed is 720 ÷ 4 = 180 hours.
180 hours = 7 days 12 hours.
Monday 7:00 a.m. + 7 days 12 hours = 7:00 p.m. on the following Monday.
Note that a gain of exactly 12 minutes (three real hours) or 24 minutes would make the face look close to right but not right: the hands only coincide when the gain is a full 12 hours.
3. Is the teacher correct that a slightly fast clock shows the correct time twice a day? Use Wednesday's 10:12 a.m. observation to test her reasoning.
Read the worked answer
The teacher is wrong, and the Wednesday reading confirms the arithmetic.
From Monday 7:00 a.m. to Wednesday 7:00 a.m. is 48 real hours, so the gain is 48 × 4 = 192 minutes = 3 h 12 min. The clock should read 7:00 + 3:12 = 10:12 a.m., which matches what she saw. So the observation is consistent with the model but shows the clock is not correct at that moment.
A correct reading requires the gain to be 0, 720, 1440, … minutes. Since the gain grows at 4 minutes per real hour, correct readings occur once every 180 real hours - about once a week, not twice a day. Her error is confusing 'the face can look plausible at many moments' with 'the displayed time equals the true time'.