Editor’s note: After introducing frontloaded progressive anagrams, which give your typical progressive anagram a couple of the perks of a perfect progressive anagram, Franklin announces the definition of a new category of branching anagram based off of perfect transgressive anagrams.

This post assumes you are familiar with the concept of transgressive anagrams (TAs), perfect TAs, and how to construct them, so ideally you will have read those two posts from The Jerx before beginning this one. In particular, it assumes you are aware that, while progressive anagrams are constructed to minimize the number of misses (“no” answers), transgressive anagrams are constructed to minimize the total number of questions.

Here is an example of a minimal transgressive anagram you can use to pin down one of the 13 official Disney princesses (up and right is No, down and right is Yes):

Of course, just because it’s minimal doesn’t mean it’s perfect. A perfect transgressive anagram would not only ask the fewest number of questions, it would also ask about the exact same sequence of letters regardless of how the participant responds.¹ The table for a perfect TA has the same letter all the way down in every column, like the N for question number 3 in the above table.

Aside from being easier to remember, this nice property gives many presentational advantages. Andy gives some of them, but here are some examples you could apply immediately:

• The letters can be written down in advance.
• The questions can be committed to a pre-recording.
• The letters can be drawn from an alphabet deck.
• The questions can be asked in any order.

Unfortunately, just like with progressive anagrams, there is no perfect transgressive anagram for most real-world categories. If you want to use one, you have to synthetically engineer a set of words that just happens to have this property. As fun as this can be, it certainly limits the use cases and presentational angles. You have to account for the unnaturalness of the set of words somehow because it will always be apparent either that the category has been artificially limited or that it doesn’t make sense as a category at all.

Fortunately for us, the nice property described above isn’t exclusive to perfect TAs. We can achieve that property for natural categories too as long as we are willing to ask a few more questions sometimes. And that brings us to the point of this post.

So You’re Going to Explain the Post’s Title Now?

Yup! Mathematicians refer to the nice property we’ve been discussing as commutativity. This post is about the construction and use of commutative anagrams.

It turns out that when you relax the “perfect” requirement that you always ask the fewest number of questions possible, you can move on to the real business of finding sets of letters that completely split up words from completely natural categories. I’ll cut right to the chase with an example:

For Disney princesses, if you know which of the letters amongst E,N,M,O, and I it does or does not contain, then you know which princess it is. I’ve drawn out the branching chart here, but the tree format is entirely obviated by this fact. If you don’t need to ask the questions in one specific order then you don’t need to memorize a tree. All you need to know is the mapping from “which of these letters it contains” to “which princess it is”. One easy way to do that would be to write down the “yes” letters as you get them in alphabetical order. Or just mentally put them together in alphabetical order. And then you just have the pairing from those short sequences to their matching answers memorized:

This is maybe slightly more than you’d need to memorize than you would for a perfect transgressive anagram, but keep in mind that the list of words itself serves as a natural mnemonic for the pairing—you know what letters are in a word because that’s how words work. Personally, when I use them, I only memorize the list of words and the list of letters, and then just reconstruct the pairing on the fly. But if you want to commit the entire pairing to memory, it is still no more to remember than the entire tree diagram you have to memorize for a typical imperfect transgressive anagram despite the added presentational flexibility it brings. Or, you know, you could just have a crib with this list concealed somewhere. Why not?

Where does this set of letters come from?

If you construct a transgressive anagram that uses as few distinct letters as possible, then that set of letters, asked in any order, is your commutative anagram.

In general, finding the smallest possible set of letters that completely distinguishes a given list is a very difficult problem. Fortunately, we’re magicians. We aren’t typically doing this with sets of 200 words. And for small problems like that, there are simple ways to guarantee you’ll get at least decently close to the best possible answer.

If you want to find such a letter set manually, use the process Andy described except that instead of independently finding a good letter to split each set of words you generate, try to find a letter that does a good job splitting all of them. At each iteration, each depth of splitting, always try to find a letter that does a good job splitting as many of those sets as possible. If you have to pick two or more different letters at a given level to achieve a good split, that’s fine, but keep track of the letters you have used and always attempt to use ones you’ve already used when splitting a set they haven’t been used on. The goal is to do exactly what Andy’s process does, but always using as few distinct letters as possible throughout the process. It may take a bit more trial and error than usual, but there’s a good chance you can find a fairly small set of letters that’s only just slightly bigger than the shortest transgressive anagram that is possible.²

Of course, reality continues to be messy, and sometimes the shortest commutative anagram is much longer than the shortest general anagram. I’ve often found it’s only longer by one or two letters, but there are no hard and fast guarantees. You just have to cross your fingers and hope for the best.

Or, you know, you could do what I’ve done and write some code that does it for you with great haste. But more on that at a future time….

1. For the purposes of future discussion, I would say that these two properties merely define a minimal commutative anagram, and a perfect transgressive anagram has the additional property that the number of words is exactly of power of two. But that’s just me being picky. ↩︎
2. And, in the rare case that your set of words does admit a perfect transgressive anagram, this process will also ensure that you find it! ↩︎