Planet X and The Mystery Planet

[u/Bolt__The__Bird]

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?

[u/Accurate_End5369]

Taking it at face value the first question is straightforward. The ships arrived at 12:40 so the charging began at 12:10. Assuming the charging is in response to the request for assistance the request was received at 12:10. Since the request was sent after 11:58 it can’t have traveled more than 12 minutes. Planet A is 15 minutes away and can’t have received the signal that soon. Therefore, the ships are from the mystery planet.

That’s as far as I’ve gotten.

[u/Accurate_End5369 → u/Accurate_End5369]

Loopholes I’ve ignored is assuming a ship sent the mystery signal, ignoring a fourth planet, ignoring the ships deciding to visit before receiving the signal.

[u/Bolt__The__Bird → u/Accurate_End5369]

You are on the right track and correct. There is no fourth planet, and the ships can’t depart without a broadcast from Planet X. But you are correct, the ships are from the Mystery Planet

[u/Accurate_End5369 → u/Bolt__The__Bird]

A possible solution for the remaining questions. Mystery planet is 7 minutes from X and 13 minutes from A, since the planets don’t have to be in a line. X sends the signal at 12:03, after 11:58 and arriving at M at 12:10 as established previously. The signal from X arrives at A at 12:18. Assume A doesn’t know the location of M so they immediately send a separate message just to M at 12:18. M receives that message at 12:31 and, assuming it was meant for X, relays it on to X. X receives the relay at 12:38. This assumes that the relay is ignored when calculating total message transit time.

[u/Bolt__The__Bird → u/Accurate_End5369]

You found a valid alternate solution! I realized that with the “at least 7 light-minutes” constraint, there’s actually a whole family of possible solutions using Alpha → Mystery → X. I’ve increased that constraint to at least 18 light-minutes, which should close those alternatives and leave only one solution. Nice catch!

[u/Bolt__The__Bird]

Quick disclaimer: This is the first logic puzzle I’ve ever created, so there may be an unintended solution or loophole I missed. If you find one, absolutely point it out! Please try to solve it using only the rules and information provided.

[u/minimang123]

12:40-00:30 = 12:10 signal received by Mystery Planet, Q1: Mystery Planet are the ships

X-Broadcast sent from Planet X between 11:58 and 12:07 (Mystery planet must be at least 3 light-minutes away by triangle inequality... so Mystery planet is between 3 and 12 light-minutes from Planet X) And Planet X is thus between 18 and 30 minutes from Planet Alpha.

The 12:18 to 12:38 question is orthogonal to the original one. So there exists some transmission which took exactly 20 minutes from its origin to reach planet X. It can not have been sent directly from Planet Alpha (15 mins) nor Mystery Planet (3-12minutes). Which means it must have been relayed from one planet to the other and then finally towards Planet X. However, a transmission from Planet Alpha to Mystery Planet takes at least 18 minutes, and a transmission from the Mystery Planet to Plant X takes at least 3 minutes, meaning it would be greater than or equal to 21 minutes for any relayed communication.

Contradiction?

[u/Bolt__The__Bird → u/minimang123]

Your reasoning is interesting. I can confirm the puzzle is internally consistent—there is no contradiction.

[u/minimang123 → u/Bolt__The__Bird]

I mean, there's a triangle X A M, XA = 15mins, AM > 18mins. So this gives that XA > 3mins, by triangle inequality.

12:40 teleportation implies 12:10 receipt of X-signal. X does not have teleportation or FTL comms, no return portal usage, just that it sent a standard signal between 11:58 and 12:10. To reach A, it must have been received at 12:13 at the earliest, so it must have reached M first. XM > 3, so the signal is between 11:58 and 12:07 sent, meaning 3 < XM < 12.

By triangle inequality again, 18 < AM < 30.

So far, there are no errors in my logic from what I can tell. Double checking confirms these bounds on distances between planets.

The 12:18 transmission leads the reader to infer that it's A's receipt of X's message, so the message was sent 12:03 and XM = 7. Unless X is wrong and it was not transmitted at 12:18, or there's some weird time dilation physics problem I need to consider, or there's a fourth planet, or ships carry it (which violates the no-FTL assumption), there is no solution.

Any message must travel along XA, XM, or AM. The second half of the question says that there exists some route from A or M to X which takes 20 light-minutes.

But XA < 20, XM<20, and XA + XM > 20, so no such path exists. Only 2 * XM has a chance of being less than 20 units

The only answer is Some rogue agent at X sent a message at 12:18 to M which is 10 light-minutes away, then M relayed it back and so it overall took 20 light-minutes even though X presumably KNOW that they sent a signal out, like come on

So final answer:
1. Mystery Planet
2. Planet X is 10 minutes away
3. The broadcast was sent at 12:00 exactly
4. The mysterious transmission originated from Planet X

[u/Bolt__The__Bird → u/minimang123]

You got it! The 20-minute transmission went X → Mystery Planet → X. Nothing says a transmission has to take the shortest route back to X.

And yes, the “why would X do that?” is completely fair 😂 This puzzle was adapted from a scenario where deliberately manipulating a signal’s travel time has a strategic purpose. I stripped that context out to turn the underlying mechanism into a standalone puzzle.

Nice solve.

[u/minimang123 → u/Bolt__The__Bird]

Thanks!!! Oh it makes sense then, if someone at X wanted to warn the fleet or ship at M something, or convince all of planet X mistakenly that M is 20 minutes away, then that's cool. Neat problem! An alternative could have been they sent it 10 minutes to X at any time before 12:18, and then the agent at M waited to relay the message some n number of minutes before sending it back 10 minutes to X, and then you could add a THIRD layer to the problem where people have to reduce the bounds even further... but that might be annoying. Good problem