Brainteasers & Logic
Clock, Angle & Number Puzzles
Overview
Clock puzzles are geometry problems in disguise — translate to angles with a simple formula and the hard part vanishes. Number puzzles about last digits or remainders are modular arithmetic: find the repeating cycle and reduce. Both reward writing a clean equation rather than staring at the problem.
How to Recognize
- →Clock hand angles at a specific time
- →'How many times do clock hands overlap in 12 hours?'
- →Last-digit and remainder puzzles
- →Self-referential number properties
Step-by-Step Approach
- 1.Clock: minute hand moves 6°/min; hour hand moves 0.5°/min
- 2.Angle between hands: |30H − 5.5M| degrees (take min with 360 minus this)
- 3.Hands overlap: every 720/11 minutes → exactly 11 times per 12-hour period
- 4.For last-digit/remainder: find the cycle length and use modular arithmetic
Key Formulas
Worked Examples
Problem
What is the angle between the clock hands at 2:20?
Solution
Hour hand at 2:20: 30×2 + 0.5×20 = 60 + 10 = 70° from 12.
Minute hand at 2:20: 6×20 = 120° from 12.
Angle = |120 − 70| = 50°. Less than 180°, so 50° is the answer.
Formula shortcut: |30H − 5.5M| = |30×2 − 5.5×20| = |60 − 110| = 50°. ✓
Answer
50 degrees. Shortcut: .
Common Mistakes
- !
Saying clock hands overlap 12 times in 12 hours — it's 11. They don't overlap every 60 minutes.
- !
Forgetting to take the smaller angle: always report min(angle, 360°−angle).
- !
Off-by-one in cycle position: if n mod cycleLength = 0, use the LAST element of the cycle, not the first.
Practice Problems
Click "Show Answer" to revealWhat is the angle between clock hands at 6:30?
What is the last digit of 9^{999}?
At what time between 3 and 4 o'clock are the hands exactly perpendicular (90°)?