Introduction
Many clock problems ask how often the hour and minute hands form a specific angle (for example 0°, 90°, or 45°) within a period (12 hours or 24 hours). Understanding the relative motion between the hands and how to solve the angle equation helps you count occurrences and find exact times.
This pattern is important because it combines algebraic roots with modular/time normalization and clarifies exceptions (when one root falls outside the hour).
Pattern: Hands Forming Same Angle Multiple Times
Pattern
Key concept: Use the relative-angle equation and root normalization to find times, then count valid roots across the time-window.
Formula to find minutes past hour H when hands make an angle θ:
|(30H - (11/2)·m)| = θ
Rearranged linear forms give the two roots per hour (before normalization):
m = (60/11)·(H ± θ/30)
Practical steps:
- Compute the two roots m for each integer hour H = 0,1,2,...,11.
- Normalize roots: accept a root if 0 ≤ m < 60; if m ≥ 60 subtract 60 and shift the hour by +1 (or interpret it as the next hour's root); if m < 0 add 60 and shift hour -1.
- Count all normalized valid times inside the requested interval (e.g., 12 hours). Note exceptions: θ = 0° (coincidence) occurs 11 times in 12 hours; θ = 180° (opposition) also yields 11 identical positions in 12 hours; many other θ give 22 occurrences (two per hour) but check normalization exceptions hour-by-hour.
Step-by-Step Example
Question
How many times do the hands of a clock form a right angle (90°) in 12 hours?
Solution
-
Step 1: Use the root formula for θ = 90°
For θ = 90°, θ/30 = 3, so the two roots per hour are:
m = (60/11)·(H ± 3) -
Step 2: Examine roots hour-by-hour
For each integer H = 0 to 11 compute m₁ = (60/11)(H - 3) and m₂ = (60/11)(H + 3).
After normalization (accepting only 0 ≤ m < 60 and shifting hours when needed) one typically gets two valid minutes in almost every hour. -
Step 3: Count valid occurrences
Right angles occur twice in most hours, giving 24 naive counts across 12 hours. However two of those naive occurrences fall outside the 0-12 hour window once each (they shift into adjacent hours when normalized), so the actual count reduces to 22.
(Equivalently: there are 2 right-angle positions per hour × 12 hours = 24, minus 2 duplicate/out-of-hour roots = 22.) -
Final Answer:
22 times -
Quick Check:
Standard result recall: right angles occur 22 times in 12 hours (and 44 in 24 hours). This matches the hour-by-hour root-check method. ✅
Quick Variations
1. Coincidence (0°): Hands coincide 11 times in 12 hours (not 12).
2. Opposition (180°): Hands are opposite 11 times in 12 hours.
3. General θ (not 0 or 180): Usually gives 22 occurrences in 12 hours - but always verify normalization for H = 0..11 because occasionally one root per hour falls outside 0-60 and shifts to adjacent hour.
Trick to Always Use
- Step 1 → Apply m = (60/11)·(H ± θ/30) for each H (0..11).
- Step 2 → Normalize each root (0 ≤ m < 60). If m ≥ 60 subtract 60 and treat as next-hour root; if m < 0 add 60 and treat as previous-hour root.
- Step 3 → Tally unique normalized times in the requested interval; watch for boundary duplicates (they reduce the naive 2×12 count).
Summary
Summary
- Key takeaway 1: Use m = (60/11)·(H ± θ/30) to get two algebraic roots per hour for angle θ.
- Key takeaway 2: Normalize roots into 0 ≤ m < 60 and shift hours when m falls outside this range.
- Key takeaway 3: Count normalized unique times across the interval; be aware that θ = 0° and θ = 180° give 11 occurrences in 12 hours (not 12).
- Key takeaway 4: When in doubt, compute hour-by-hour and perform a quick check against known standard results (e.g., 90° → 22 times in 12 hours).
Example to remember:
Right angle (90°) → 22 times in 12 hours (two per hour except for two shifted roots).
Hint: Except 0°/180°, most angles appear 22 times in 12 hours.
Common Mistakes: Assuming 2×12 = 24 without normalization adjustments.