Connections

[boharr]

Here's a question for any of you math wizards who play the NYT game Connections. Four sets in four rows. Sixteen squares.
My daughter said to me that once you get a first row solved correctly, the game gets easier. Well, yes. But then I wondered if there is a way to determine mathematicly how much easier it gets as you solve. Of course, once you have three rows picked successfully, the last row is a 100 percenter.

Note: there are no intentional red herrings in this query.

Thanks!

[MMe]

Not a math wizard, but there is simple way to determine those odds if you mean just guessing randomly.

If there are 16 left, if you pick a random tile, the chance of the next one matching is 3/15, then 2/14, then 1/13, so the odds of randomly getting 4 in a row at the start are 6/(15*14*13), or one in 455.
If there are 12 left, similarly, your odds are 6/(11*10*9), or one in 165.
With 8 left, it's 6/(7*6*5), or one in 35.
And with 4 left, as you say you are in pretty good shape.

[boharr]

Simple, hey? Thanks!

[DBMiller]

Another math way to look at it: There are 16c4 ways to choose 4 items from 16 objects. That's 1820, of which there are only 4 correct groupings, so 1820/4 = 455, just as MMe got. (You can type "16 choose 4" into google, and it'll tell you that it's 1820)

12c4 = 495, 495/3 = 165. Again, same as MMe. Now as to how much easier the game just got? You had a 1/455 for the first set, versus 1/165 for the second set. So the game got about two-thirds easier (455-165) / 455 = 63.74% (You've eliminated 290 of those first 455 chances).

8c4 = 70, 70/2 = 35. and the game got (165-35) / 165 = 78.8% easier.

And obviously it's 100% chance to get it right once you have the first three. Can't get any easier than that!