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,^{9} 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

From Al Crouch, a little bit of mathematical history:

The following set of number pairs refers to some coincidences in American history, spanning the period from the founding of the nation in 1776 to the present.

2, 6

?, ?

9, 23

17, 36

26, 32

What number pair should be in the second position?

## Riddler Classic

From Benjamin Danard, a collegiate constitutional:

You are a professor at a university. Each day you walk from one building to another through a square courtyard that is 50 feet by 50 feet. You enter the courtyard from the center of the west wall and exit from the center of the south wall. You don’t like sharp turns, so you make sure your turns have at least a 5-foot radius. You also don’t like to cross your own path when walking.

Here’s an example of how you might walk across the courtyard:

One day while walking this courtyard you begin to wonder: What is the *longest* path you could take?

*Extra credit*: You want your walk to be some certain length (say, 350 feet). Can you calculate a path of that length through the courtyard?

## Solution to last week’s Riddler Express

Congratulations to Daniel Elegant of Naperville, Illinois, winner of last week’s Riddler Express!

Last week brought us Riddler Nation’s first ever mystery story, which had to do with you, a very wealthy individual, being kidnapped, marooned on a small desert island, and abandoned with little more than some sandwiches and a satellite phone. (I refer you to last week’s column for the full and somewhat lengthy story.) Your challenge was to determine what you should say when you called your people to help them locate you.

This puzzle’s co-submitter, Mark Baird, has the life-saving answer:

You call your people, knowing that what you say will be communicated to the best person to receive it, at *exactly* daybreak. You tell them, “I’m OK, it’s precisely sunrise. I’ll call again at sunset.” At exactly sunset, you make the second call, saying, “It’s exactly sunset, and I’m still OK.” You even have a few seconds left on the battery for contingencies, but none should arise.

Your call at sunrise locates you along the great circle between the poles at the penumbra — the space between shadow and light. You don’t know what time it is, but your people do, so the line of penumbra at that moment constitutes a line of longitude between the poles of globe illumination. Your second call provides a crucial piece of information: how long daylight lasted where you are, which fixes your latitude.

Had you been kidnapped at equinox season, when daylight has the same duration the world over, finding your latitude would be more involved, requiring, for example, finding the ratio between a shadow’s length and the height of a water bottle at noon, and identifying the hemisphere by which side of the bottle the shadow extends from (north or south).

## Solution to last week’s Riddler Classic

Congratulations to François Maillot of London, winner of last week’s Riddler Classic!

Last week brought further international intrigue with the challenge faced by the fictional Dr. Lana Gurtin, a mathematician hired by British intelligence in 1942. The Germans were rolling out a new tank, the Uberpanzer, each of which was stamped with a serial number according to when it came off the line — so the first one built was numbered 1, the second one 2, etc. A number of the tanks were spotted by British scouts, who dutifully recorded their numbers. They sent this data to intelligence headquarters. However, it was intercepted along the way by a German spy, who destroyed *most* of the data. By the time the British agents caught up to the spy, they could recover only two pieces of information. First, the lowest serial number recorded was 22. Second, the highest serial number recorded was 114. Given that, what was Dr. Gurtin’s best estimate of the total number of Uberpanzers the Germans had built?

She estimates that there are **135 tanks**.

Theodore James, this puzzle’s submitter, explains why:

This is a variant of the classic German tank problem, which goes as follows: Set T is a set of consecutive integers from 1 to N, where N is not known. Give a random sample (with no duplicates) from T with k observations, what is the best method for estimating N? The “standard” answer to this problem is M + M/k – 1, where M is the maximum value of the random sample. This is intuitive — the answer is simply the maximum value plus the average distance between observations, i.e. M + (M-k)/k = M + M/k – 1.

However, in last week’s riddle, we were denied k — we didn’t know the size of the set of observations. Our best bet, then, was to use L – 1, where L is the minimum value of the set of observations, to estimate the average distance between observations. This gives us M + L – 1 = 114 + 22 – 1 = 135.

Solver Steve Langasek put it this way: “If the tanks are a random sample, then the most likely distribution is that the lowest serial number is as close to the beginning of the sequence (No. 1) as the highest number is to the end of the sequence. So for a *single* guess, this is what the good doctor would guess.”

Solver Marc Lischka explained it like this: “Due to symmetry, the distribution of ‘lowest serial number recorded’ is the same as the distribution of (2 × ‘average serial number’ – ‘highest serial number recorded’). Thus, the sum (‘lowest serial number recorded’ + ‘highest serial number recorded’) has an expected value of (2 × ‘average serial number’), which is (‘number of tanks’ + 1). Hence, ‘number of tanks’ is estimated to be (‘lowest serial number recorded’ + ‘highest serial number recorded’ – 1).”

Finally, solver Laurent Lessard provided a useful discussion of biased versus unbiased estimators — with an application to German tanks, of course — and the tradeoff between precision and accuracy.

## 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 me at [email protected]