Our Most Controversial Problem Revamped

Almost exactly a year ago, I posed a card puzzle here that sparked a frenzy in the comments section. The solution to that puzzle was so counterintuitive that several readers rejected it outright. In honor of the approaching anniversary, it’s time to stir up more controversy with two more surprising card puzzles.

One of these I learned from Presh Talwalkar and the other from Martin Gardner. So please forward any hate mail their way.

Did you miss last week’s puzzle? Check it out here, and find its solution at the bottom of today’s article. Be careful not to read too far ahead if you haven’t solved last week’s yet!

Puzzle #46: Coming Up Aces

1. Shuffle a normal face-down deck of 52 playing cards and then flip one card at a time face up.

Which card is more likely to come immediately after the first ace appears: the king of spades or the ace of spades? In other words, you’ll flip over cards until you see an ace of any suit. Is the next card more likely to be the king of spades or the ace of spades, or do they have equal likelihood?

2. Reshuffle the same deck and start flipping again. This time, before flipping, you must guess when the first black ace will appear. Which position in the deck is most likely, or are they all the same?

I’ll be back Monday with the answers and a new puzzle. Do you know a cool puzzle that you think should be featured here? Message me on X @JackPMurtagh or email me at gizmodopuzzle@gmail.com

Solution to Puzzle #45: There’s No Place Like Home

Last week’s puzzle asked you to slip into the role of a sports statistician. Did you figure out how scheduling affects the NBA championship series?

Shout-out to adanarg13 for your well-articulated answer.

In real basketball, you could argue that the order of the games matters for psychological reasons (e.g., enjoying a streak might increase your chance of winning the next game), but in our mathematical model it turns out the order of the games doesn’t matter at all. In both cases, the Celtics would still be more likely to win simply because they have one extra home game, even if they end up not using it. We’ll argue for the best-of-seven case, but the same argument extends to other numbers of games.

The key insight is that, even if a team wins four games before game seven, then playing out the rest of the games just for fun could not change the outcome of the championship. So we can view the series as though they always play seven games no matter what and then look at who won more games after all seven have been played. The team that plays at home more often has an edge.