Maths olympiad question · P5–P6★★★★ Expert
Clocks and time: olympiad question 8
An expert question on clocks and time for P5–P6, with a step-by-step solution, diagrams and the answer. Every year of the Hua Xia Cup has clock questions, from time passed in P1 to the angle between the hands in P6.
The question
A clock with hands keeps perfect time.
From 3:00 am to 1:20 pm the same day (the first and last moments included), how many times are the two hands mirror images of each other in the line through 12 and 6?
(Both hands lying together on that line counts too.)
How many times?
Show the hints (3)
- Hint 1: Give each hand's position as the angle clockwise from 12. When the hands are mirrored in the line through 12 and 6, how are the two angles related?
- Hint 2: When mirrored, hour angle + minute angle = 360° (or 720°, …). The minute hand turns 6° a minute and the hour hand 1/2°: how fast does the sum grow?
- Hint 3: The sum grows from 90° to 4120° (13/2° a minute). Count the multiples of 360 in between.
Worked solution
Have a go first, then read the solution step by step.
- 1.Measure angles clockwise from 12. Mirrored in the line through 12 and 6, the hands make equal angles with it on opposite sides, so hour angle + minute angle = 360° (plus whole turns).
- 2.At the start (3:00 am): minute hand 0°, hour hand 90 + 0 ÷ 2 = 90°, sum 90°.
- 3.The sum grows 6 + 1/2 = 13/2° a minute. To 1:20 pm is 620 minutes: 620 × 13 ÷ 2 = 4030°, so the sum goes from 90° to 4120°.
- 4.In between, ends included: 360°, 720°, … 3960°, (3960 − 360) ÷ 360 + 1 = 11 times.
- ⚠️Common mistake: assuming 11 times every 12 hours, as for the hands meeting — meeting is about the difference of the angles (11/2° a minute); mirroring is about their sum (13/2° a minute), 13 times in 12 hours. And check whether the first or last moment is exactly one.
Answer: 11 times
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