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The Mysterious Benedict Society:
Mr. Benedict's Book of Perplexing Puzzles, Elusive Enigmas, and Curious Conundrums
v.
The Real World


The Book

The Mysterious Benedict Society: Mr. Benedict's Book of Perplexing Puzzles, Elusive Enigmas, and Curious Conundrums
By Trenton Lee Stuart, Illustrated by Diana Sudyka
Published by Little, Brown and Company. First Edition: October 2011.
ISBN 978-0-316-18193-8


Problematic Puzzles, Erroneous Enigmas, and Confusing Conundrums


That's Far Enough (page 30)

It is exactly 11:45 AM, and Mr. Curtain has caught up with you at last. In fact, his wheelchair is now only sixty feet away from where you are securely tied up.

Much to his chagrin, however, Mr. Curtain's wheelchair malfunctions. When he tries to move forward sixty feet, his wheelchair goes only half that far. Of course, this means he is now only thirty feet away. He zooms forward, but again only half as fas as he would like . . . which means he is now merely fifteen feet from where you are a sitting duck.

In this manner, Mr. Curtain keeps lurching forward, each time getting closer. But with every lurch, he covers only half the distance than remains between you.

At what time will Mr. Curtain's wheelchair crash into you?

__ __ __ __ __

["N E V E R" is apparently the expected answer.]


This is a version of one of Zeno's paradoxes, and the Hint for it (page 141), while unclear, suggests a very unsatisfactory analysis:

If you keep dividing a number in half, will you ever reach zero?
Being able to break a task down into an infinite number of steps is not the same as proving that that task requires an infinite amount of time to complete. The literature (more than two-thousand years' worth) abounds with explanations of why the former does not imply the latter. Moving any object from one point to another suffers from the same "difficulty", but ordinary motion does seem to be possible in the real world.

In this particular case, the problem statement is sufficiently ambiguous that almost any result is possible. We have no idea, for example, how the speed of the chair depends on time. How does the chair accelerate when it starts to move? If it stops, how does it decelerate to a stop? Or should we pretend that it behaves in some unspecified, idealized/simplified way, in which a real chair (with non-zero mass) can not behave?

Presumably, "his wheelchair goes only half that far" means that it stops after it has gone "half that far", but for how long does it remain stopped? Can Mr. Curtain restart it instantaneously, or must it stay at rest for some non-zero time interval? After a few jerky (or smooth -- we can't tell) attempts to move, he might well learn to anticipate the stops, and instantly reactivate the chair. The time he spends stopped at each attempt to move does not even need to be zero. If the stopped time is only shrinking as fast as the time he spends moving, then "NEVER" is the wrong answer.

The mathematics of infinite sequences and series like these might not be familiar, even to exceptional children, so it might have been better to omit this kind of problem, tempting as it may have been, rather than to include such a sloppy example with such a misleading "Hint".

Also, while far from the biggest blemish in this problem, a starting distance of sixty-four feet rather than sixty feet would have allowed more even divisions by two. (Thirty-two meters might have been an even better choice.)


Moocho's Perfect Pie (page 35)

This problem asks us to divide a (circular) pie among nine people. (Apparently, the baker himself gets none.)

If you required only two pieces of pie, you could simply make a single cut down the middle. But if everyone who wants a piece is to enjoy one, what is the smallest number of cuts you must make?
What, exactly, does "cut" mean here? Straight cut? Any-shape-I-please cut? Edge to edge, or anywhere to anywhere? One gracefully curving cut alone can yield any reasonable number of pieces. As with "That's Far Enough", the number of spaces provided for the answer (four) implies (or at least hints at) what the expected answer is, but, as with "That's Far Enough", the problem statement here is so ambiguous that the actual answer is not well defined.

Extra Credit: While four straight, edge-to-edge cuts can yield nine pieces, how many other piece counts can be obtained using four straight, edge-to-edge cuts?

Hint: Consider simpler problems first. How many pieces can be obtained using zero straight, edge-to-edge cuts? One cut? Two cuts? Three cuts? (Follow this link to see some related pictures.)


The Ten Men (page 76)

[...] a laser pointer that fires a real laser, [...]
Incredible as it may seem, a "laser pointer" is called a "laser pointer" precisely because it employs a laser, which might also be called a "real laser". The laser in a typical laser pointer is not a particularly powerful laser, but the difference is entirely quantitative, not in any way qualitative. Again, literature on this topic abounds.

A small, weak laser and a large, powerful laser are both lasers. A laser is a laser.


Salamander vs. Whisperer (pages 78, 79)

Original Salamander vs. Whisperer diagram

For quantity and quality of defects, this one wins the prize. The original diagram is reproduced at the right.

Near a high reflective wall, you succeed in taking control of Mr. Curtain's Salamander and discover that it has been modified with a special transmitter. This transmitter emits a spherical energy field capable of disabling any electrical device.

Your mission is to disable the Whisperer, which is positioned nearby.

Where should you aim the transmitter on the reflective wall in order to disable the Whisperer?



Viewed from its center, a sphere is the same in all directions, so, if the "transmitter emits a spherical energy field", then it doesn't matter which way you point the thing, so why aim it? (As the Pointless Man once said, "A point in every direction is the same as no point at all.") But let's ignore this fatal error in the problem statement, and pretend that we actually have a transmitter which emits a narrow beam, one which is susceptible to being aimed.

In the original diagram, there is no obvious barrier between the Salamander and the Whisperer, so why would any rational person waste the time and effort needed to calculate a bank shot off a reflective wall, when a direct shot seems eminently practical? (Indeed, with a "spherical energy field", quite unavoidable.) But let's ignore this fatal error in the problem statement, and pretend that we actually need to use the reflective wall.

The ray-gun-like transmitter on the Salamander seems obvious enough, but where, exactly, is "the Whisperer"? That is, at what, exactly, are we supposed to aim? The helmet? The chair? Where? In the original diagram, the Whisperer is so large that nothing like a precise aim seems possible.

The Hint (page 143) suggests an understanding of how a mirror works which is confused, at best:

Draw a ninety-degree angle that emanates from the wall. One of its rays must meet the end of the transmitter, while the other ray meets the Whisperer.

"Draw a ninety-degree angle [...]." Why? Mirrors do not always reflect at ninety-degree angles. If they did, then finding one's face in the morning would be a much greater challenge than it is. Assuming that a ninety-degree angle would appear anywhere in a problem like this is asking for an express ticket to the wrong answer.

The negative numbers in the number line along the reflective wall were an interesting touch (especially quirky with Up being more negative), but seem more likely to cause confusion than to be helpful in any way. What might have been helpful would have been a diagram on a quadrille-ruled background ("graph paper"), where the weapon's "muzzle" and the target point (whatever it might be) were clearly identified, and situated at nice points on the rulings. The diagram should also have included rulings on the far side of the reflective wall, with enough room to reflect the position of either the target or the weapon's "muzzle" over to the other side of the wall. For example:

Benedict Reflect Layout
(Click on the diagram to get a clean image, suitable for printing.)

Here,

In this layout (unlike in the original diagram), every object has a well-defined position, and a Barrier (B) blocks any direct shot. For convenience, the coordinate system was chosen to place the reflective surface of the Reflective wall at horizontal coordinate 0. The vertical coordinates are entirely arbitrary, with 0 at no special location. (Some negative values were included, for old times' sake.)

Solving the problem still requires knowing how a mirror works in the real world. Follow this link to see a solution.

One might wonder what the expected (not really the "right") answer was. On the bright side, this answer (whatever it might be) contributes nothing to the book's ultimate goal (decoding the secret message), so the damage caused by this mess is only local.

Or perhaps this was one of those intentionally ambiguous ("without-setting-foot-on-a-blue-or-black-square") problem statements, so what look like glaring errors are actually bits of cleverness, and anyone who didn't realize this missed the point completely. Seems unlikely, though.


Sticky's Glossary, "transfer protocol" (page 139)

transfer protocol, n. also known as hypertext transfer protocol, is a universal standard for addressing websites on the Internet, including www.mysteriousbenedictsociety.com

The well-read Sticky must know better than this.

The HyperText Transfer Protocol (HTTP) is a transfer protocol, not the transfer protocol. The File Transfer Protocol (FTP) and the Simple Mail Transfer Protocol (SMTP) are other, older transfer protocols. Other transfer protocols abound, although not all of them have acronyms which include "TP". (The Internet Message Access Protocol (IMAP) and the Post Office Protocol (POP) are two examples of these which deal with e-mail.)

A transfer protocol describes a system of formats and procedures used to exchange messages between computers (or even between parts of a single computer). It is definitely not "a universal standard for addressing websites on the Internet." Here, a "message" might be a request, such as a request to accept an e-mail message, or a request to send a Web page (like "GET / HTTP/1.1"), or it might be a response to a request, such as the actual Web page data, or some kind of error message (like "404 Not Found").

A Uniform Resource Identifier (URI) or Uniform Resource Locator (URL) is a standard way to identify a resource (such as a document -- for example, a Web page) on the Internet.

A name like "www.mysteriousbenedictsociety.com" is a Domain Name System (DNS) symbolic name, which is used instead of an Internet Protocol (IP) address to specify a Web site. One could call DNS a "standard for addressing websites on the Internet", but that's not a complete description of DNS, and DNS is not a "transfer protocol".

In a URI like "http://www.mysteriousbenedictsociety.com/news.html", "http" specifies the "scheme" (in this case, HTTP, that is, the protocol used to request and transmit the resource in question); "www.mysteriousbenedictsociety.com" specifies the computer offering the document (currently at IP address 206.34.121.249); and "news.html" specifies the resource (in this case a particular Web page in HyperText Markup Language (HTML)).

In fact, strictly speaking, because it lacks a scheme, the example provided, "www.mysteriousbenedictsociety.com", is rather ambiguous, and hence not a very good example of a URI. Nowadays, almost any Web browser will add "http://" when no scheme is specified, but, while a complete URI like "http://www.mysteriousbenedictsociety.com/" does specify a Web page, a bare domain name like "www.mysteriousbenedictsociety.com" does not.

Extra Credit: Type "www.mysteriousbenedictsociety.com" into the address window of a Web browser, and hit Enter, Go, Return, or whatever you need to hit to make something happen. Now look at what the Web browser has in its address window. That (most likely) is a URI.

It would be nice if every complex topic had a nice, simple explanation. Sadly, too much simplifying tends to produce an explanation which may be nice and simple, but as untrue as it is simple. Many things hide behind a Web browser's pretty face. Trying to cram an accurate explanation of the World-Wide Web, the Internet, Internet transfer protocols, the HyperText Transfer Protocol, the Domain Name System, and everything else back there into one short sentence is probably doomed to fail.

A Web search for any of the technical terms used here should lead to any number of more detailed explanations of those terms, almost any of which would be more helpful than Sticky's explanation of "transfer protocol".


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