So, what kinds of mathematical expressions are Leta, AKA have a value equal to the number of letters they have? We’ve got the most basic two possible, “four” and “0” – the single word, and the single numeral. These are unique Leta expressions – there’s no other single word or single numeral that will work here. What other categories could we explore?
Expressions that Evaluate to 0
Next, we have a whole family of simple Leta expressions: almost any kind of purely numerical arithmetic that evaluates to 0. For example, “5 – 5”, or “32 + 42 – 52”, or “999999 * 0”. As long as we don’t use any letters in the expression, and you end up with 0, every single one of these expressions (and there’s infinitely many of them) are Leta. They’re not the most interesting Leta expressions, but they definitely count.
Simple Addition
Now, the first set of Leta expressions that might be more interesting are based on word arithmetic. Let’s start with the easiest of them all, simple addition: are there any phrases that look like ___ + ___ that work?
A quick try with some everyday numbers gives us a few working options. “ONE + SEVEN” is Leta, as is “TWO + FIVE” and a few others. But, is there a way we can come up with a systematic way to describe every phrase that works? Maybe we can assign a score to each number that tells us how it can be used in one of these simple addition phrases.
So, let’s define a Leta score of a number.
Leta score: the value of a number minus the number of letters in that number.
For example, the Leta score of SIX is 3. SIX has a value of 6 and a letter count of 3, so the score is 6 – 3 = 3.
Similarly, the Leta score of FIVE is 1, the Leta score of ONE is -2, and the Leta score of TEN is 7. Now, I’m going to write out the first few Leta scores; let's see if that gives us any insights.
Immediately, there’s some interesting things going on here.
- Four is the only number here with a Leta score of 0! It has the same number of letters as its value, which is why it works as a standalone answer, and why the original riddle is elegant in the first place.
- The two pairs I wrote out before, “ONE + SEVEN” and “TWO + FIVE”, have Leta scores that add up to zero. SEVEN needs two extra letters to be Leta, and ONE provides those letters – and it’s the same with TWO and FIVE.
- Other combinations that add up to a Leta score of 0, like "TWO + TWO + SEVEN", or “ONE + THREE + FIVE + SIX” will also work – all we have to do to make something Leta is make sure the scores add up to 0!
- Numbers increase far faster than the words for the numbers get longer, on average. All the numbers listed after FOUR have positive Leta scores, and with a little thought, we can see that pretty much everything after that will also be positive, and getting larger for the most part.
- This means that among the natural numbers that we’re looking at, ZERO, ONE, TWO, and THREE are the only numbers with a negative score! Because we need the Leta scores to add up to zero, that means that all Leta expressions that look like “_____ + _____” have ZERO, ONE, TWO, or THREE in them! If they don’t have any of these four numbers, the sum of their scores must be positive!
So that’s another very interesting, non-trivial family of Leta expressions. Using our handy table of Leta scores, we can generate an infinite number of Leta expressions that look like “_____ + _____ + …”! And, we know a lot about what these expressions look like! Next time, we’ll try addition with more complicated things, like negative numbers!