Problem #40 EASY
The Stuck Clock
Apple Adobe Pattern Recognition Logic
Problem Statement
A wall clock stops working and gets stuck displaying a fixed time. A forgetful office worker glances at this broken clock twice each day at random moments. Remarkably, over many days of checking, the broken clock shows the correct time on roughly 2 occasions per day on average. How is this possible, and what does it reveal about how often a stopped clock is accidentally correct?
Answer & Quick Explanation
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It is simply true. A stopped 12-hour clock matches the real time exactly twice every 24 hours — once during AM and once during PM. The saying 'a broken clock is right twice a day' is a literal mathematical fact, not just a figure of speech.
Detailed Editorial Solution
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Think about what a 12-hour clock displays. It cycles through the same 12-hour sequence twice in a full day. A fixed display therefore matches the actual time at exactly two moments each day — one per 12-hour cycle.
Step 1: Suppose the broken clock is stuck at 3:47. The actual time passes through 3:47 AM and again through 3:47 PM every single day.
Step 2: At each of those two moments, the broken clock's display exactly matches the real time.
Step 3: In every 24-hour period, there are exactly 2 instances where a stopped 12-hour clock is correct.
Step 4: The office worker checks at random moments. Over many days, by chance, some of those random glances happen to land at or very near the two correct moments.
Step 5: This is not about probability — it is a deterministic fact. Every stopped 12-hour clock is correct exactly twice per day, every day, forever.
Step 6: A 24-hour clock (military time display) stopped at a fixed time would be correct exactly once per day.
Key Insight:
The humour and surprise of this riddle comes from treating 'a broken clock is right twice a day' as just a saying, when it is actually a precise mathematical statement. The 12-hour display format, by design, maps two different real-world times to the same visual display.