What Are Your Chances Of Winning The U.S. Open?

Welcome to The Riddler. Every week for the last four years, Oliver Roeder has offered up problems related to the things we hold dear around here: math, logic and probability. I will do my best to continue that legacy as The Riddler’s new editor.

As you may know, two puzzles are presented each week: the Riddler Express for those of you who want something bite-size and the Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,1 and you may get a shoutout in next week’s column. If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter.

## Riddler Express

You and your friend are playing a game of “Acchi, Muite, Hoi” (or what some on Twitter have been calling the “lookaway challenge”). In the first round, you point in one of four directions: up, down, left or right. At the exact same time, your friend also looks in one of those four directions. If your friend looks in the same direction you’re pointing, you win! Otherwise, you switch roles as the game continues to the next round — now your friend points in a direction and you try to look away. As long as no one wins, you keep switching off who points and who looks.

It seems like your chances of winning should be 50 percent, since there are exactly two players. But surely it’s not that simple. If both you and your friend choose your directions randomly in each round, what are your chances of winning?

## Riddler Classic

The U.S. Open is upon us! Dave Moran and Erich Friedman each submitted a tennis-themed puzzle that asked what it really takes to succeed in the tournament.

Suppose you’re playing a match at the U.S. Open, and you’re slightly better than the competition: your chances of winning any given point are exactly 55 percent. (Yes, most players are more likely to win the points they serve, but we’re simplifying things a bit.) What are your chances of winning a three-set match, as played by the women, or a five-set match, as played by the men? And what are your chances of winning the whole tournament (seven consecutive matches)?

If you’re not familiar with the scoring system in tennis, you can read more about it here. In short, the first to 4 points wins a game (as long as they’ve won 2 more points than their opponent), the first to six games wins a set (as long as they’ve won two more games), and the first to two sets (for women) or three sets (for men) wins the match. If at any point a set is tied at six games apiece, that set is decided by a tiebreaker, in which the first to 7 points wins (with the same 2-point margin rule applying here).

You know, I thought tennis scoring was pretty straightforward until I had to write it down.

## Solution to last week’s Riddler

At long last, Riddler Nation has a new warlord! All hail 👑 David Love 👑 of Ambler, Pennsylvania, winner of last week’s Riddler!

Longtime puzzle editor Oliver Roeder’s tenure concluded with the fourth (but certainly not final) installment of the Riddler Nation Battle Royale. Here is how he described the rules: In a distant, war-torn land, there were 10 castles. There were two warlords: you and your archenemy. Each castle had its own strategic value for a would-be conqueror. Specifically, the castles were worth 1, 2, 3, …, 9 and 10 victory points. You and your enemy each had 100 soldiers to distribute, in any way you like, to fight at any of the 10 castles. Whoever sent more soldiers to a given castle conquered that castle and won its victory points. (If you each sent the same number of troops, you split the points.) You didn’t know the distribution of the enemy forces until the battles began. Whoever won the most points won the war.

You submitted a plan distributing your 100 soldiers among the 10 castles. I took all these battle plans and adjudicated every possible one-on-one matchup.

I received more than 1,000 battle plans, of which more than 900 were valid.2 Here’s a graphic that shows the strategies of all those entries:

Each row is a different warlord (the best warlords are at the top, the worst are at the bottom), and each column is a different castle (1 is on the left, 10 is on the right). Castles with more soldiers are shaded darker. The 10 castles are collectively worth 55 points, so to win a match you needed to collect at least 28 points. At a glance, you can see that strategies where soldiers were more evenly distributed didn’t fare as well, while the strongest warlords focused their efforts on collecting exactly 28 points in each match. Sure enough, the best warlords placed most of their soldiers on castles 4, 5, 9, and 10, which together are worth 28 points.

Here were the top performers:

##### Riddler Nation’s strongest warlords (and Vince Vatter)
soldiers per Castle Record
Name 1 2 3 4 5 6 7 8 9 10 W T L
1 David Love 0 0 0 12 19 2 3 3 35 26 774 4 153
2 Benjamin Choi 0 0 0 13 17 1 1 1 31 36 772 6 153
3 Brayden Kooistra 1 1 1 15 17 1 1 1 30 32 770 2 159
4 Russell McClellan 0 1 1 15 18 1 3 2 35 24 765 2 164
5 Bill Neagle 0 0 10 1 1 23 3 3 34 25 758 9 164
18 Vince Vatter 0 0 11 1 2 23 4 2 34 23 739 15 177

As noted in the table, former two-time top warlord Vince Vatter did well, but not well enough to retain his title. I’m sure Vince will exact revenge in the next battle.

There’s a lot of data to digest here, so I hope all you aspiring warlords learn from our winners! If you have ideas for tweaking the rules to mix things up (e.g., 13 castles instead of 10, as Vince has suggested), let me know. Surely, Riddler Nation won’t have a second two-time champion…

## Want more riddles?

Well, aren’t you lucky? There’s a whole book full of the best puzzles from this column and some never-before-seen head-scratchers. It’s called “The Riddler,” and it’s in stores now!

## Want to submit a riddle?

Email Zach Wissner-Gross at riddlercolumn@gmail.com.

## Footnotes

1. Important small print: For you to be eligible, I need to receive your correct answer before 11:59 p.m. Eastern time on Monday. Have a great weekend!

2. Meaning they only used 100 soldiers, only whole numbers of soldiers and so forth.

Zach Wissner-Gross leads development of math curriculum at Amplify Education and is FiveThirtyEight’s Riddler editor.