Hands Apart: 2:20
Read the puzzle
An eight-sided clock face with the numbers 12, 3, 6 and 9 and tick marks for every hour shows twenty past two; a digital display underneath reads 2:20. The minute hand points at the 4, and the hour hand sits just past the 2. What is the smaller angle between the two hands?
What is the SMALLER angle between the hands?
In 20 minutes the hour hand moves 10° past the 2.
50°
Minute hand at 120°, hour hand at 60° + 10° = 70°: 50° apart.
How to solve it
Place each hand in degrees, measured clockwise from the 12. Each hour gap is 30°, the minute hand moves 6° per minute and the hour hand moves 0.5° per minute.
The minute hand at 20 minutes is at 20 × 6 = 120°, on the 4.
The hour hand is at 2 × 30 = 60° for two full hours plus 20 × 0.5 = 10° for the twenty minutes, so it sits at 70°, a third of the way from the 2 to the 3.
Subtract: 120 − 70 = 50°. The other way round is 360 − 50 = 310°, so the smaller angle is 50°.
The usual wrong answer is 60°, from counting two hour gaps between the 2 and the 4. That treats the hour hand as frozen on the 2, but in 20 minutes it has already moved 10° toward the minute hand, shrinking the gap. Check with |30 × H − 5.5 × M| = |60 − 110| = 50°.
