Why a “1 in a million” New York Times Connections solution is counterintuitively likely

After I solved the New York Times Connections puzzle today, the “Connections Bot” told me that my solution was unique among the last million players. The same thing happened yesterday. How can I repeatedly be getting one-in-a-million solutions? The answer reveals a little about our mistaken intuition about probability.

The 10 trillion ways to attempt to solve a Connections puzzle

Connections asks you to divide sixteen words or phrases into four sometimes vexing categories. Here’s today’s puzzle:

As usual, there are fake leads in here. You might wonder what goes with ROCK, PAPER, and SCISSORS — but if you did, you’d already be off on the wrong track.

So, how many ways are there to solve a puzzle like this?

There’s only one paragraph of probability theory in my explanation, below, and you can skip it if you want.

Selecting four items from a list of sixteen is a typical probability problem. It uses the mathematical idea of a factorial, denoted with an exclamation point: n! is the product of all the numbers from 1 to n. The number of combinations of a group of n items taken from a collection of p items is a fraction where the numerator is n! and the denominator is p! times (n-p)!

Luckily both Google Sheets and Microsoft Excel have a very convenient function COMBIN(p,n) that will calculate that for you.

In this case, the relevant fact is that on your first turn, you are selecting four items out of sixteen (or as the statisticians would say, “16 choose 4”). COMBIN(16,4) rapidly scalculates that there are 1,820 possible ways to select four items in the first move.

If your first move is wrong, there are still 1,819 other guesses you could make. In fact, assuming you don’t make any right guesses, through four turns the number of possible guesses is 1,820 x 1,819 x 1,818 x 1,817, which is 10.93 trillion possibilities. (I’d put an exclamation point on that sentence but I already used up my quota of exclamation points explaining factorials.)

That’s almost, but not quite, the actual number of possible ways to play the first four turns of Connections. If you guess correctly, you remove four items from the list of items you can guess, and you’ll only be picking four out of twelve on your next turn. Once you guess right again, you’ll be down to picking four out of eight. I’ve done the math for you, and it turns out, through four turns, there are:

• 10.8 trillion ways to guess wrong four times.
• 32.6 billion ways to guess right once and wrong three times.
• 55.0 million ways to guess right twice and wrong twice.
• 57.0 thousand ways to guess right three times and wrong once.
• 24 ways to get the puzzle right with no errors. (There are four ways to pick the first correct group, three to pick the second, two to pick the third, and of course the fourth group is just the remaining items, no guessing required. 4 x 3 x 2 = 24 ways to solve the puzzle.)

I’m ignoring solutions that take more than four guesses, because they don’t add much to our understanding but they do make the calculations more complex. They don’t affect the rest of the analysis much.

Some startling conclusions from this analysis

Here’s what you can learn from these numbers.

First off, if you guess at random with no actual knowledge of the words involved, your chances of solving the puzzle in four guesses are microscopic. You have only a one in 334 chance of getting even one right answer. Your chance of guessing the whole puzzle by tapping boxes at random is one in 453 billion.

Let’s look at that one-in-a-million number. Suppose for the sake of argument that a million people attempted to solve the puzzle just by clicking boxes at random. If you do the same, your chance of matching one of the other million people is about one in ten million. You’re a virtual lock to get a unique answer and see the “1 in 1 Million” message on the Connections Bot.

Of course, people don’t actually guess at random. It’s an intellectual puzzle, not a roulette wheel. If you guess all the categories right, you probably did that the same way as tens of thousands of other people. (Since there are 24 ways to get the answer with no mistakes, it’s no surprise if the Connections Bot tells you one in 24, or even one in ten, people solved the puzzle the same way you did.)

Even if you make a mistaken guess, it’s likely that you made the same wrong guess as lots of other people. There are often red herrings in the puzzle, like the Rock/Paper/Scissors in today’s puzzle. If your knowledge of professional wrestlers or rappers or 18th century poets isn’t complete, you’re probably going to go wrong in the same ways as lots of other folks.

But if you make an oddball guess or start clicking boxes at random because your idea of the possible category doesn’t match anybody else’s, you could easily end up in a spot where nobody else has gone. With more than 10 trillion possible combinations of guesses, it’s pretty easy to do that.

That’s where my one-in-a-million results came from, and probably yours, too. It turns out that if you go sufficiently astray, a one-in-a-million result is actually quite likely.

That might offend your intuition about probability. But that’s what happens when you’re tromping around in a 10-trillion possibility space of guesses.