Opposite directions (“pointing towards each other”) in 24 hours — How many times do the hour and minute hands face each other (180° apart) in a day?

Introduction / Context:“Pointing towards each other” is typically interpreted as the hands being in opposite directions, i.e., separated by 180°. Like overlaps, these opposite positions follow a repeating pattern tied to the relative speed of the hands.

Given Data / Assumptions:

• Relative speed = 5.5° per minute.
• Opposite positions correspond to an angular difference of 180° (or equivalently −180°).
• The pattern over any 12-hour span repeats in the next 12 hours.

Concept / Approach:In 12 hours, opposite positions occur 11 times (analogous to 11 overlaps). Therefore, in 24 hours, the count doubles.

Step-by-Step Solution:Opposites in 12 hours = 11.Opposites in 24 hours = 2 × 11 = 22.

Verification / Alternative check:General formula within an hour interval: t (minutes after H:00) satisfies |30H − 5.5t| = 180. Solving across H = 0..10 yields 11 solutions per 12-hour cycle (excluding boundary duplication).

Why Other Options Are Wrong:24 assumes two per hour, which is incorrect; 20 and 12 undercount the standard result.

Common Pitfalls:Mixing up counts for right angles (22 per 12 h) with opposites (11 per 12 h).

Final Answer:22