# Infinite Wordle and the mastermind numbers, CUNY Logic Workshop, March 2022

This will be an in-person talk for the CUNY Logic Workshop at the Graduate Center of the City University of New York on 11 March 2022.

Abstract. I shall introduce and consider the natural infinitary variations of Wordle, Absurdle, and Mastermind. Infinite Wordle extends the familiar finite game to infinite words and transfinite play—the code-breaker aims to discover a hidden codeword selected from a dictionary $\Delta\subseteq\Sigma^\omega$ of infinite words over a countable alphabet $\Sigma$ by making a sequence of successive guesswords, receiving feedback after each guess concerning its accuracy. For any dictionary using the usual 26-letter alphabet, for example, the code-breaker can win in at most 26 guesses, and more generally in $n$ guesses for alphabets of finite size $n$. Meanwhile, for some dictionaries on an infinite alphabet, infinite play is required, but the code-breaker can always win by stage $\omega$ on a countable alphabet, for any fixed dictionary. Infinite Mastermind, in contrast, is a subtler game than Wordle because only the number and not the position of correct bits is given. When duplication of colors is allowed, nevertheless, the code-breaker can still always win by stage $\omega$, but in the no-duplication variation, no countable number of guesses (even transfinite) is sufficient for the code-breaker to win. I therefore introduce the mastermind number, denoted $\frak{mm}$, defined to be the size of the smallest winning no-duplication Mastermind guessing set, a new cardinal characteristic of the continuum, which I prove is bounded below by the additivity number $\text{add}(\mathcal{M})$ of the meager ideal and bounded above by the covering number $\text{cov}(\mathcal{M})$. In particular, the precise value of the mastermind number is independent of ZFC and can consistently be strictly between $\aleph_1$ and the continuum $2^{\aleph_0}$. In simplified Mastermind, where the feedback given at each stage includes only the numbers of correct and incorrect bits (omitting information about rearrangements), then the corresponding simplified mastermind number is exactly the eventually different number $\frak{d}(\neq^*)$.

I am preparing an article on the topic, which will be available soon.

## 6 thoughts on “Infinite Wordle and the mastermind numbers, CUNY Logic Workshop, March 2022”

1. Are drop ins allowed to come to CUNY on 3/11 to hear this lecture? If so, what time and what address/room?

• I shall find out and post back here. In the midst of Covid, there were access rules at CUNY, but I am unsure whether those are still in effect.

2. My wife and I were trying to figure out what the solution was to the (finite portion of the) infinite wordle that appears above. That 3-letter word with a t in the middle is very perplexing, if there is supposed to be a sentence structure. I was thinking “eta” because of the “tao” Greek letter connection, but that’s ruled out.

• Ah, perhaps I should have had a cleverer plan with that example. In the general case, the “dictionary” can be quite arbitrary, and that is all I had meant to suggest. Can you help me design a more enlightening example? I’m basically flexible as to word length, but it should be long, and it would be good to use supercalifragilistic and antidisestablishmentarianism or other well-known very long words, perhaps something German? I’d like the bottom row to be something clever, with the guesses making sense as the play proceeds, and pleasing color pattern, with gradually more greens.

• I actually really liked your first four lines. I wouldn’t change those at all (except, of course, the coloring of some tiles, if you decide to change the solution). If you want to make the solution more mathematical, you could make it “one fish tao isnt eta fish…”. Alternatively, “one fish tao is a big fish in a mathematical pond…” (speaking of the Fields medalist).