Unbreakable One-time Pads
Enigmatic and mysterious numbers stations broadcasts are weird to the point of being spine-tingling. What makes them perhaps even more compelling is that they're still in use, decades after the world wars that sparked their rise. They've survived the communications revolutions because unlike so much digitized code, these transmissions are unbreakable.
You may scoff at the idea. If codes generated by supercomputers can be hacked, what's so hard about deciphering a few numbers? It is hard. Actually, it's nearly impossible. Done properly, no one will understand the message but the intended recipient.
It works because of the one-time pad concept. In this system, both the sender and receiver have a single copy of paper (or other media) filled with random digits or numbers. Using a key that corresponds to the pad, the recipient can figure out what letters the incoming characters or digits stand for.
Crucially, each key is used only once, thus the name one-time pad. Both the sender and receiver immediately destroy the key at the end of the broadcast, meaning that even if someone else intercepts the series of numbers, they'll have no way to decode the hidden message.
This also means that there's no relationship between past or future messages, so there's no pattern for cryptographers to use in breaking the code. Each new key and message are randomly generated, and done properly, the system is the only way to perfectly encrypt a transmission.
To fool the ingenuity of the best cryptographers, however, the key and message must be truly random. And as it turns out, creating really random sequence is challenging, even for computers. Sans complete randomization, there may be a pattern for experts to detect, and they can crack at least part of the message. Sadly for codebreakers, because of the one-time pad concept, their hard work won't help them decipher future messages.
To make things more confounding, stations may send mostly dummy broadcasts that mean nothing at all. This forces adversaries to expend far more resources attempting to figure out which messages are real and which are fake.
Once in a while the broadcasts are less than professional. The person reading the code may make mistakes, or there could be laughing in the background. Cuban numbers stations in particular are known for their hilarious missteps -- sometimes they have such poor transmission that they're impossible to hear, or they accidentally pipe music from Radio Havana onto the airwaves.