đźš¨đźš¨đźš¨ **â€śThe Riddlerâ€ť book**** is out now!** Itâ€™s chock-full of the best puzzles from this column (and, fret not, their answers) and some riddles that have never been seen before. I hope you enjoy it, and thank you for riddling with us these past three years. đźš¨đźš¨đźš¨

Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. There are two types: Riddler Express for those of you who want something bite-size and 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 weekâ€™s column. If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter.

## Riddler Express

From Philip Ruo, a puzzle of pondering playing perfection:

The NFL season is in full swing, and only one undefeated team remains â€” the 6-0 Los Angeles Rams. In theory, though, given the current NFL scheduling scheme â€” or at least what Wikipedia says it is â€” what is the largest number of teams that *could* finish a regular season 16-0?

## Riddler Classic

From Ricky Jacobson and Ben Holtz, geological disaster looms beneath:

You live on the volcanic archipelago of Riddleria. The Riddlerian Islands are a 30-minute boat ride off the shores of the nearby mainland. Your archipelago is connected via a network of bridges, forming one unified community. In an effort to conserve resources, the ancient Riddlerians who built this network opted not to build bridges between any two islands that were already connected to the community otherwise. Hence, there is exactly one path from any one island to any other island.

One day, you feel the ground start to rumble â€” the islandsâ€™ volcanoes are stirring. Youâ€™re not sure whether any volcano is going to blow, but you and the rest of the Riddlerians flee the archipelago in rowboats bound for the mainland just to be safe. But as you leave, you look back and wonder what will become of your home.

Each island contains exactly one volcano. You know that if a volcano erupts, the subterranean pressure change will be so great that the volcano will collapse in on itself, causing its island â€” and any connected bridges â€” to crumble into the ocean. Remarkably, other islands will be spared unless their own volcanoes erupt. But if enough bridges go down, your once-unified archipelagic community could split into several smaller, disjointed communities.

If there were *N* islands in the archipelago originally and each volcano erupts independently with probability *p*, how many disjointed communities can you expect to find when you return? What value of *p* maximizes this number?

## Solution to last weekâ€™s Riddler Express

Congratulations to đź‘Ź Sarry Al-Turk đź‘Ź of Toronto, winner of last weekâ€™s Riddler Express!

Last week, you learned about a girl who loves to sing â€śThe Unbirthday Songâ€ť to people â€” as one does. But she can only do that, of course, if itâ€™s not the personâ€™s actual birthday. If she kept singing it to random people until it happened to be someoneâ€™s birthday, how long would her singing streak go before it became more likely than not that she would encounter someone whose birthday it is?

Care for those vocal cords, child: Itâ€™s **252 people**.

The probability that itâ€™s *not* an individual personâ€™s birthday is 364/365. The probability that you sing to N people in a row without it having been anyoneâ€™s birthday is \((364/365)^N\) â€” since the birthdays are independent events, we can multiply that fraction over and over to match the number of people. We want to find the number N such that that probability is larger than 0.5.

We could just stick that straight into a computer solver, but itâ€™s Friday, so letâ€™s have some fun and do a little algebra.^{2} First, we can take the logarithm of both sides, to get N out of the exponent, and then we can rearrange things a little bit:

\begin{equation*}(364/365)^N > 0.5\end{equation*}

\begin{equation*}N\log(364/365) > \log(0.5)\end{equation*}

\begin{equation*}N > \log(0.5)/\log(364/365)\end{equation*}

That numerator, log(0.5), equals about -0.3, and that denominator, log(364/365), equals about -0.001. That fraction equals about 253. So the singing streak is expected to go 252 people before it became more likely than not that a birthday boy or girl would be encountered.

And a very merry unbirthday to *you*, dear reader â€” unless, of course, it *is* the big day.

## Solution to last weekâ€™s Riddler Classic

Congratulations to đź‘Ź Clemens Fiedler đź‘Ź of Krems, Austria, winner of last weekâ€™s Riddler Classic!

Last week, a farmer wanted to tether a goat â€” as one does. Specifically, the farmer wanted to tether the goat to the fence that surrounded his circular field such that the goat could graze on exactly half the field, by area. The field had a radius R. How long should the goatâ€™s tether be?

The tether should be a bit longer than the radius â€” specifically, it should have a length of about **1.159R**.

Solver Russell No-last-name-given showed us what this looks like, pictured below. The field is green with radius R; the goatâ€™s would-be grazing area is gray, with radius r.

The picture is all fine and good, but what about that math? It looks a little nasty. And it is a little nasty, Iâ€™m afraid. But, hey, itâ€™s nearly Halloween, so letâ€™s surrender to the nastiness and fear.

The area on the field available to the goat is the intersection between two circles. And there just happens to be a whole body of knowledge about such circle-circle intersections. One way to think about it is to consider the two different shapes that the goatâ€™s tether allows it to graze in. The first is the big â€śpizza sliceâ€ť, defined in the image above by the two diagonal green radii and the bottom arc of the gray circle, and the second are the narrower areas between the pizza slice and the fence.

Hector Pefo broke this math down a little bit more for us in his diagram (this time, the goat is on the southern end of the field):

And finally, Laurent Lessard showed us a way to get to this same solution with calculus.

Either way, I hope you enjoy it, goat. Nom nom nom.

## Want more riddles?

Well, arenâ€™t you lucky? Thereâ€™s a whole book full of the best of them and some never seen before, called â€śThe Riddler,â€ť and itâ€™s in stores now!

## Want to submit a riddle?

Email me at oliver.roeder@fivethirtyeight.com.