Clocks and Calendars problems involve calculating time intervals, angles between clock hands, gain or loss of time, leap years, odd days, and determining the day of the week for any given date.

1. Clock Fundamentals and Hand Speeds

A circular clock dial is divided into $360^\circ$ and 60 minute spaces.

• Minute Hand Speed: Moves $360^\circ$ in 60 minutes = $6^\circ$ per minute.
• Hour Hand Speed: Moves $30^\circ$ in 60 minutes = $0.5^\circ$ per minute.
• Relative Speed: Minute hand gains $6^\circ - 0.5^\circ = 5.5^\circ$ per minute over the hour hand.
• Angle Between Hands Formula:Angle $\theta = |30H - 5.5M|$, where $H$ is the hour ($1$ to $12$) and $M$ is the minute ($0$ to $59$). If $\theta > 180^\circ$, the reflex angle is $360^\circ - \theta$.

Important Clock Alignments

• Coincidence ($0^\circ$ angle): Hands coincide 11 times in 12 hours and 22 times in 24 hours (due to overlap between 11 and 1).
• Opposite ($180^\circ$ angle): Hands point in opposite directions 11 times in 12 hours and 22 times in 24 hours.
• Right Angles ($90^\circ$ angle): Hands are perpendicular 22 times in 12 hours and 44 times in 24 hours.
• Straight Line ($0^\circ$ or $180^\circ$): Hands form a straight line 22 times in 12 hours and 44 times in 24 hours.

2. Calendar Fundamentals and Odd Days

The concept of odd days is central to calendar calculations. Odd days are the remainder days left after dividing total days by 7.

• Ordinary Year: 365 days = 52 weeks + 1 day = 1 odd day.
• Leap Year: 366 days = 52 weeks + 2 days = 2 odd days.
• Leap Year Rules: Non-century years are leap years if divisible by 4. Century years (100, 200, 1900, 2100) are leap years ONLY if divisible by 400 (e.g., 1600, 2000).

Odd Days in Century Blocks

• 100 years = 76 ordinary years + 24 leap years = $76(1) + 24(2) = 124$ days = 5 odd days.
• 200 years = $5 \times 2 = 10 \equiv 3$ odd days.
• 300 years = $5 \times 3 = 15 \equiv 1$ odd day.
• 400 years = $5 \times 4 + 1$ (extra leap day) $= 21 \equiv 0$ odd days.

Day Mapping by Odd Days

0 = Sunday, 1 = Monday, 2 = Tuesday, 3 = Wednesday, 4 = Thursday, 5 = Friday, 6 = Saturday.

A reliable step-by-step approach

1. For clock angles, plug $H$ and $M$ into $\theta = |30H - 5.5M|$.
2. For time when hands coincide or form an angle, set $\theta$ and solve for $M$.
3. For calendar date finding, break down the target year into completed 400-year blocks, remaining centuries, completed years, and completed months of the current year.
4. Sum all odd days, divide by 7, and match the remainder to the day code.

Common mistakes

• Counting century years like 1900 or 2100 as leap years simply because they are divisible by 4.
• Forgetting that hands overlap 22 times in 24 hours, not 24 times.
• Including Feb 29 for a year when the target date is before Feb 29.
• Taking the reflex angle when the question asks for the acute interior angle between clock hands.

Quick revision summary

• Clock Angle = $|30H - 5.5M|$.
• Relative speed = $5.5^\circ$/min.
• Ordinary year = 1 odd day; Leap year = 2 odd days.
• 100 years = 5 odd days; 400 years = 0 odd days.