Leta III: PLUS and Negative Numbers

Before we get to anything complicated like subtraction, we should cover every facet of addition we can. We’ve got a blueprint for any expression that looks like “_____ + _____ + _____...” if the blanks are natural numbers, but what about cases where they’re other kinds of numbers, or other ways to write symbols?

PLUS vs. +

We’ve been writing “+” in our Leta expressions, but there’s no reason why the + sign (which isn’t a letter) can’t be written as PLUS. The word PLUS contributes -4 to the Leta Score of an expression – we can think of it as a ZERO (no value, four letters) that needs to be put in between any pair of words. This doesn’t complicate the expressions too much, but now we’re looking for sums of two words that have a Leta Score of 4, or sums of three words that have a Leta Score of 8, and so on.

Some examples of these are “SEVEN PLUS SEVEN”, “FIVE PLUS SIX”, “TWO PLUS THIRTEEN”, “FOUR PLUS EIGHT PLUS NINE”, and “SIX PLUS SIX PLUS SIX PLUS SIX”. As you can see, it’s the same as the earlier case, but you have to take into account how many terms you’re adding together.

An interesting little sidenote here, before we move into anything more complicated: when we were working only with “+” and not “PLUS”, all Leta expressions had to have a ZERO, ONE, TWO, or THREE in them, because those numbers were the only sources of negative Leta score. If you use “PLUS”, there’s a guaranteed source of negative score between each number, so you can have expressions like SEVEN PLUS SEVEN” and “FIVE PLUS SIX”. Now, is there a new set of numbers that Leta expressions must have at least one of? Well, let’s extend our table of Leta scores a bit.

 

 

It's pretty clear that the scores are just going to keep increasing as we look at higher and higher numbers, so there must be a point at which any expression made of only larger numbers can’t have be Leta. It looks like NINE is that point – all numbers greater than NINE have Leta scores that are 5 or higher. A PLUS can only contribute -4 per number (and there will always be a number left over), so there can’t be a Leta expression that looks like “_____ PLUS _____ PLUS _____ PLUS ….” where all the blanks are NINE or more. So our theoretical maximum is EIGHT – all Leta expressions of this type must have at least one ZERO, ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, or EIGHT. Can we get this maximum lower, to SEVEN? No, because I can think of a counterexample that works with only numbers EIGHT or higher. “EIGHT PLUS EIGHT PLUS EIGHT PLUS EIGHT” works. So, we know that every Leta expression in this form must have a number that’s less than NINE!

 

Negative Numbers

The word NEGATIVE contributes -8 to the Leta Score of an expression. So, as you get more and more negative, the scores leap negative quickly, with NEGATIVE ONE having a Leta Score of -1 – 11 = -12. Let’s look at the first few negative numbers:

 

 

 

Because every Leta score for a negative number is of the form (negative number) – (positive number), we know that they’ll always be negative. So we can use the same rule as before to generalize a statement for all Leta expressions that look like “_____ + _____ + _____ + …”. Any of these are possible as long as the Leta Scores add up to zero.

Here’s a few more interesting expressions that are Leta, now that we can use negative numbers. “NEGATIVE ONE + TEN + THIRTEEN”, “NEGATIVE ONE + TWENTY-ONE”, “NEGATIVE TWO PLUS TWENTY-SIX”, and “FOUR HUNDRED AND TWENTY-FOUR + NEGATIVE THREE HUNDRED AND SEVENTY”.

And now that we’re dealing with negative numbers, there’s no analogue to our nice “all Leta Expressions must have these numbers” limitation. Any number, no matter how arbitrarily high, can be countered with an arbitrarily low number, and vice versa.

That’s enough for this time, but as we’ve only scratched the surface so far!

Leta II: Base Cases and Simple Addition

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!

The Number of Letters (Leta I)

One of my absolute favorite puzzles from when I was a child – and honestly, maybe the puzzle that got me into puzzles in the first place – was this simple phrase in a book that I’ve forgotten the name of.

 

“The answer is the number of letters in the answer.”

 

 I’ll give you a chance to solve it before I discuss it more – it’s really elegant and worth a thought, I think.

 

 ~ ~ ~ ~ ~

 

The given answer is four. Four is the only word, in English, that has the same number of letters as its value. That’s a very interesting and universal fact that this puzzle is built around, and thinking about it and solving it leads you naturally to that fact! That’s what sets this puzzle apart from many other, more contrived puzzles.

But there is another answer that was suggested by the book: 0. Because 0 is a numeral and not a letter, it also contains the same number of letters as its value. Which brings me to an interesting question… what else has this quality? Surely lots of different phrases and mathematical expressions work, and there’s a tapestry of answers instead of just “four” and “0”!

All that’s left is to give this problem a terrible name, just like the rest of em. It’s about a number or expression having a self-referential or meta number of letters… meta letters – Leta. That’s appropriately bad, I think. If an expression has a number of letters equal to its value, call it Leta. So, next time, let’s try to investigate what kinds of expressions are Leta!

Birthday Puzzle 2024

I make a puzzle every year on my birthday, always trying to make something brand new that pushes some personal limit. This year, I've gone with something, um, substantially less accessible than puzzles I've made in the past, but it "needed to be made" more than any of my other puzzles.

There is the concept of a "PokeDoku", which is a Pokemon-inspired take on a Sudoku. It consists of a 3x3 grid that must be filled out with 9 Pokemon total, and each row and column shares a characteristic, like "Fire-Type" or "Mega Pokemon". I found these puzzles very unsatisfying - there are many answers for each spot, and your score is determined by how few players chose that particular answer. Also, the categories are arbitrary, and it's only 3x3, where Sudoku is a full, perfectly balanced 9x9. So, I set out to try and make a better PokeDoku, and the result is below.

Now, if you happen to be one of the few people who identify as both a puzzle fiend and a Pokemaniac, you may be able to solve this puzzle without any kind of direction, so I'll just give you the puzzle and the most basic instructions.

• Your goal is to fill out this 6x6 grid with unique Pokemon such that each row, column, and colored 2x2 square has something different in common. (These are big, obvious characteristics and not something like "They both have a height between 4'11" and 6'1" ".)
  
• All variants of Pokemon present in the main series games can used, using their in-game names. Temporary form changes are associated with the base Pokemon, but are still considered their own Pokemon (with their own names).
• There are a few very similar sets of Pokemon that will fill this grid, but there is only one unique solution when it comes to the row, column, and square characteristics.
  

 

But I do want to give a bit more direction to make this puzzle accessible outside of the hardcore Pokemaniacs out there, so read on for those details.

• Columns in the puzzle represent Pokemon that share a type. All columns have a unique type.
  
• Rows in the puzzle represent Pokemon that start with the same letter of the alphabet. All rows have a unique letter.
  
• Squares in the puzzle represent the Pokemon's generation in order, starting from Generation I at the top left and ending with Generation IX at the bottom right.
• Regional variations of Pokemon go by the same name, and therefore start with the same letter. Alolan Rattata is referred to as Rattata in-game, and so counts for R, not A.
• Temporary form changes of Pokemon do have a different in-game name: Mega Pidgeot counts for M, not P. However, the temporary forms are assigned the generation of their base form, not the generation they were officially revealed.
• As mentioned above, there are a few cases in which multiple Pokemon can fill a single square (these Pokemon would have to share a type, a generation, and the first letter of their names). But there are only six types and six starting letters that are possible, providing a unique solution up to symmetry.

 You should have everything you need to solve the puzzle! It should be completely solvable without any guesswork.

Happy solving! I might do a breakdown of the puzzle here in a month or so, because hoo boy, it was a problem with too many constraints.