Three sources activate every 4, 6 and 9 minutes. At 12:00 the three of them coincided. A visitor arrives between 12:00 and 12:30 and observes that: the first and second will match again in 3 minutes; the second and third will match again in 6 minutes. What time did you arrive?

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Synchronized sources

Synchronized sources work when the reader accepts a simple rule and follows it to the end without losing the thread. That's where the beautiful part appears: in the clarity of the mechanism.

Three sources activate every 4, 6 and 9 minutes. At 12:00 the three of them coincided. A visitor arrives between 12:00 and 12:30 and observes that:

the first and second will match again in 3 minutes;
the second and third will match again in 6 minutes. What time did you arrive?

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Hints

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It first lists all times between 12:00 and 12:30 when two sources match exactly.
It then translates 'first match from two sources occurs 2 minutes after entry' to a candidate entry time.
Also check when the next triple match drops to lock in a single entry.

Solution

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Answer: came in at 12:16. The three sources each agree:

$$ \operatorname{mcm}(4,6,9)=36\ \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\text{minutos}. $$ If they matched at 12:00, the next triple match is at 12:36. Since that triple match happened 20 minutes after entering: $$

12{:}36-20\ \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\text{min} = 12{:}16.
$$ Match checking from exactly two sources as of 12:16: - at 2 min: 12:18 (6 and 9 matches),

at 8 min: 12:24 (4 and 6 matches),
at 20 min: 12:36 (3 or'clock matches).

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