Will You Top the Leaderboard?

You’re halfway through your workout on your stationary bike, when you notice your ranking on the leaderboard. Where can you expect to rank by the end of your workout?

Welcome to Fiddler on the Proof! The Fiddler is the spiritual successor to FiveThirtyEight’s The Riddler column, which ran for eight years under the stewardship of myself and Ollie Roeder.

Each week, I present mathematical puzzles intended to both challenge and delight you. Beyond these, I also hope to share occasional writings about the broader mathematical and puzzle communities.

Puzzles come out Friday mornings (8 a.m. Eastern time). Most can be solved with careful thought, pencil and paper, and the aid of a calculator. Many include “extra credit,” where the analysis gets particularly hairy or where you might turn to a computer for assistance.

I’ll also give a shoutout to 🎻 one lucky winner 🎻 of the previous week’s puzzle, chosen randomly from among those who submit their solution before 11:59 p.m. the Monday after that puzzle was released. I’ll do my best to read through all the submissions and give additional shoutouts to creative approaches or awesome visualizations, the latter of which could receive 🎬 Best Picture Awards 🎬.

This Week’s Fiddler

You’re doing a 30-minute workout on your stationary bike. There’s a live leaderboard that tracks your progress, along with the progress of everyone else who is currently riding, measured in units of energy called kilojoules. (For reference, one kilojoule is 1000 Watt-seconds.) Once someone completes their ride, they are removed from the leaderboard.

Suppose many riders are doing the 30-minute workout right now, and that they all begin at random times, with many starting before you and many starting after. Further suppose that they are burning kilojoules at different constant rates (i.e., everyone is riding at constant power) that are uniformly distributed between 0 and 200 Watts.

Halfway through (i.e., 15 minutes into) your workout, you notice that you’re exactly halfway up the leaderboard. How far up the leaderboard can you expect to be as you’re finishing your workout?

As an added bonus problem (though not quite Extra Credit), what’s the highest up the leaderboard you could expect to be 15 minutes into your workout?

This Week’s Extra Credit

Again, suppose there are many riders starting their 30-minute workouts at random times, and that their powers are uniformly distributed between 0 and 200 Watts. Now, suppose you decide that you too will be pedaling with a random (but constant) power between 0 and 200 Watts.

If you look down at the leaderboard at a random time during this random workout, how far up the leaderboard can you expect to be, on average?

Making the Rounds

There’s so much more puzzling goodness out there, I’d be remiss if I didn’t share some of it here. This week, I’m sharing that longtime reader Xavier Durawa has started a brand-new Substack you might enjoy (given that you’re here).

It’s called X's Puzzle Corner, and this week’s puzzle involves two oracles: one who predicts how stocks will do over the next day, and another who predicts how stocks will do over the next hour. (This has me wondering if there’s some way to leverage the predictions of both oracles to earn even more money.)

Want to Submit a Puzzle Idea?

Then do it! Your puzzle could be the highlight of everyone’s weekend. If you have a puzzle idea, shoot me an email. I love it when ideas also come with solutions, but that’s not a requirement.

Last Week’s Fiddler

Congratulations to the (randomly selected) winner from last week: 🎻 Andy Quick 🎻 from Kitchener-Waterloo, Canada. I received 80 timely submissions, of which 75 were correct—good for a 94 percent solve rate, and the greatest number of solvers since this past February. Nice job, everyone!

Last week, you considered the game of TENZI. In the original game, each player had 10 dice of the same color. To get started, you rolled all 10 dice—whichever number came up most frequently became your target number. In the event multiple numbers came up most frequently, you could choose your target number from among them. At this point, you put aside all the dice that came up with your target number.

From there, you continued rolling any remaining dice, putting aside any that come up with your target number. Once all 10 dice showed the same number, you yelled, “Tenzi!” If you were the first to do so, you won.

Last week, you specifically explored a simplified version of the game in which you began with three total dice (call it “THREEZI”?) rather than 10.

On average, how many dice did you put aside after first rolling all three?

With three dice, the solvers known as the “MassMutual Fiddlers” recognized  there were three distinct possibilities:

• The same number came up for all three dice. In this case, you put all three dice aside (and presumably yelled, “Threezi!”).

• The same number came up for two dice, while a different number came up for the third die. In this case, you put two dice aside.

• A different number came up for each die. In this case, you put aside one die.

Your task was to determine the respective probabilities of these three possibilities, and use them to calculate the expected value for the number of dice you put aside.

First, let’s look at getting the same number on all three dice. There were six ways this could happen (they could all come up 1, 2, 3, 4, 5, or 6) among the 6³, or 216, total ways of rolling the dice. That meant the probability of this event was 6/216, or 1/36.

Next, let’s look at getting the same number on exactly two dice. There were 6 ways to pick which number that was, 5 remaining ways to pick the number for the third die, and 3 ways to pick which two dice were the same (the first and second, the first and third, or the second and third). That meant the probability of this event was (6·5·3)/216, or 5/12.

You could find the probability of the third and final case by subtracting the two aforementioned probabilities from 1, but let’s calculate it directly instead. There were 6 ways to pick the value for the first die, 5 remaining values for the second die, and 4 remaining values for the third. That meant the probability of this event was (6·5·4)/216, or 5/9.

Sure enough, these three probabilities—1/36, 5/12, and 5/9—summed to 1, suggesting all of our calculations up to this point have been correct.

To recap, the probability you put aside three dice was 1/36, the probability you put aside two dice was 5/12, and the probability you put aside one die was 5/9. Therefore, the expected (or average) number of dice you put aside was 1/36·3 + 5/12·2 + 5/9·1, which came to 53/36, or about 1.472. That was slightly less than one-and-a-half dice, or about halfway to “Threezi!”

Last Week’s Extra Credit

Congratulations to the (randomly selected) winner from last week: 🎻 Andrew Ford 🎻 from Madison, Wisconsin. I received 34 timely submissions, of which 31 were correct—good for a 91 percent solve rate.

For Extra Credit, we returned to the original game of TENZI, which had 10 dice.

On average, how many dice did you put aside after first rolling all 10?