Chain letters are often illegal, break their promises, and are super annoying. Also, they are immortal. At least, that's the only way we can think to explain the persistence of the appealing but mathematically impossible "secret sister" hoax, which started popping up on Facebook last month and shows no signs of stopping, despite more than a century of attempts to debunk letters and emails like it.

The "secret sister" scheme promises a bounty of gifts worth hundreds of dollars for anyone who passes along the Facebook post (or email) as instructed: Take the list of names you get, send a gift worth \$10 to the person at the top. Remove their name from the top of the list, add yours to the bottom, and send to six of your friends with the same instructions. The idea is, eventually, your name will be at the top, and a bunch of strangers will give you free stuff.

In other words, it's a chain letter, and a pyramid scheme. Both have been around for a long time, and as long as they've circulated, there have been mathematical models to debunk them.

There are a couple of reasons why, as appealing as they sound, these schemes don't really work.

First, not everyone who gets a letter participates, meaning that no matter how much you like the six people you mail that secret sister list to with your name on it, chances are some will end up in the trash. And how do we know that? Because if everyone who got one of these things participated, an unimaginable number of people would be sending letters and gifts through the mail in just weeks.

Based on models looking at past iterations of those schemes, we can get a look at how the secret sister might play out.

There's at least one problem with chain letters. They're illegal if they request money or other items of value and promise a substantial return to the participants.

There seem to be a few variations on how these exchanges are precisely worded, but assuming that the list you're on is six people long, and the instructions require that participants add their names to the bottom of the list, and mail out their part within a week, it would take six weeks - and 46,656 participants - before your name moves up to the top into the gift-receiving spot. That's a lot of people to rely on for an unbroken chain, and it gets worse. In 13 weeks, if everyone participated, the number of people involved would be more than there are on Earth. It's a pyramid scheme, and although sometimes they work as promised for those who create them or are involved very early on, pyramid schemes quickly require an impossible number of participants in order to avoid collapse.

Also, as Snopes noted in their debunking of this particular chain, the secret sister chain is likely illegal. Here's what the U.S. Postal Service says about the practice:

"There's at least one problem with chain letters. They're illegal if they request money or other items of value and promise a substantial return to the participants. Chain letters are a form of gambling, and sending them through the mail (or delivering them in person or by computer, but mailing money to participate) violates Title 18, United States Code, Section 1302, the Postal Lottery Statute. (Chain letters that ask for items of minor value, like picture postcards or recipes, may be mailed, since such items are not things of value within the meaning of the law.)"

Chain letters like these - promising luck, financial returns, or simply the perpetuation of a charitable spirit - have been around for a long, long time, as a cursory glimpse at Our Nation's Newspaper Archives makes clear. In a 1917 edition of the Evening Current in Carlsbad, New Mexico, an announcement cautioned that "No Chain-Letter Project has Red Cross Approval."

And a column that ran widely in 1922 bemoaned the "Chain Letter Epidemic," desperately urging Americans to refuse to cooperate in a "good luck" chain. That column noted that even if the good luck chain was true and passed along honestly, then every American in the country would soon have extraordinary "good luck," rendering the entire idea meaningless.