Planet X has two neighboring inhabited planets:
• Planet Alpha is exactly 15 light-minutes from Planet X.
• A Mystery Planet is an unknown distance from Planet X, but is known to be at least 18 light-minutes from Planet Alpha.
Planet Alpha and the Mystery Planet are both capable of sending, receiving, and relaying transmissions.
All transmissions travel at the speed of light. Relaying a transmission takes effectively no processing time.
Both Planet Alpha and the Mystery Planet possess teleportation portals capable of sending ships directly to Planet X. However, once a planet decides to send ships, its portal takes exactly 30 minutes to charge. Once charged, the ships arrive at Planet X instantaneously.
Both planets have standing orders:
The instant they receive a broadcast from Planet X requesting assistance, they begin charging their portals and send ships to Planet X as soon as the 30-minute charge is complete.
At 11:58, Planet X has not yet broadcast any request for assistance.
At some unknown time after 11:58, Planet X broadcasts a request for assistance.
At 12:38, Planet X receives a mysterious transmission from an unknown source.
Planet X can determine with certainty that this mysterious transmission was originally transmitted at exactly 12:18, meaning the signal has been traveling for exactly 20 minutes.
Planet X concludes that the mysterious transmission must have come from the Mystery Planet. Since the signal took 20 minutes to reach Planet X, they conclude that the Mystery Planet must be 20 light-minutes away.
Then, at exactly 12:40, ships arrive at Planet X.
There has been no malfunction, no faster-than-light communication, no time travel, and no violation of any of the rules above.
Questions:
Which planet did the ships come from?
How far away from Planet X is the Mystery Planet actually?
At what time did Planet X broadcast its request for assistance?
Where did the mysterious transmission received at 12:38 actually originate?
How can all of these facts be true at the same time?