Line Work

A problem proposed by C. Gebhardt in the Fall 1966 issue of Pi Mu Epsilon Journal:

A particular set of dominoes has 21 tiles: (1, 1), (1, 2), … (1, 6), (2, 2), … (6,6). Is it possible to lay all 21 tiles in a line so that each adjacent pair of tile ends matches (that is, each 1 abuts a 1, and so on)?

Click for Answer


fc books

Just a reminder — Futility Closet books make great gifts for people who are impossible to buy gifts for. Both contain hundreds of hand-picked favorites from our archive of curiosities. Some reviews:

“A wild, wonderful, and educational romp through history, science, zany patents, math puzzles, wonderful words (like boanthropy, hallelujatic, and andabatarian), the Devil’s Game, self-contradicting words, and so much more. Buy this book and feed your mind!” — Clifford A. Pickover, author of The Mathematics Devotional

“Futility Closet delivers concentrated doses of weird, wonderful, brain-stimulating ideas and anecdotes, curated mainly from forgotten old books. I’m hooked — there’s nothing quite like it!” — Mark Frauenfelder, founder, Boing Boing

“Meant to be read in pieces, but impossible to put down.” — Gary Antonick, editor, New York Times Numberplay blog

“Futility Closet is a dusty museum back room where one can spend minutes or hours among seldom-seen curiosities, and feel that none of the time was wasted.” — Alan Bellows,

Both books are available now on Amazon. Thanks for your support!

The Meigs Elevated Railway

In 1873, Captain J.V. Meigs patented a surprisingly advanced steam-powered monorail that he hoped could serve Boston. It followed a pair of rails set one above the other, thus requiring only a single line of supports, and it burned anthracite, to reduce smoke in city streets.

Each cylindrical car, shaped to reduce wind resistance, contained 52 revolving seats and was completely upholstered. Engineer Francis Galloupe wrote, “If it were ever desirable, one would become more easily reconciled to rolling down an embankment in one of these cars than in that of any other known form, for the entire absence of sharp corners and salient points is noticeable.”

A 227-foot demonstration line in East Cambridge carried thousands of curious riders 14 feet above Bridge Street at up to 20 mph, but in 1887 a fire, possibly started by a competing streetcar business, destroyed most of Meigs’ car shed. He wrote, “‘the most magnificent car ever built’ was melted down by the furnace into which it was thrust. Its metal plates were melted down and the little wood and upholstering burned out.” He fought on for a few more years, ran out of money, and quit.

Here’s his 1887 description of the project.


peirce ink drop

A drop of ink has fallen upon the paper and I have walled it round. Now every point of the area within the walls is either black or white; and no point is both black and white. That is plain. The black is, however, all in one spot or blot; it is within bounds. There is a line of demarcation between the black and the white. Now I ask about the points of this line, are they black or white? Why one more than the other? Are they (A) both black and white or (B) neither black nor white? Why A more than B, or B more than A? It is certainly true,

First, that every point of the area is either black or white,

Second, that no point is both black and white,

Third, that the points of the boundary are no more white than black, and no more black than white.

The logical conclusion from these three propositions is that the points of the boundary do not exist.

— Charles Sanders Peirce, “The Logic of Quantity,” 1893

A Circle Theorem
Image: Wikimedia Commons

If two chords of a circle intersect in a particular angle at S, then the sum of the opposite arc lengths (say, AB + CD) remains the same regardless of the position of S within the circle.

(Nick Lord and David Wells, “A Circular Tour of Some Circle Theorems,” Mathematical Gazette 73:465 [1989], 188-191.)

A Bureaucracy Maze

At a Mensa gathering in 2003, Robert Abbott tried out a new type of maze — five bureaucrats sit at desks, and solvers carry forms among them:

When you enter the maze you are given a form that says, ‘Take this to the desk labeled Human Resources.’ You look for the desk with the nameplate Human Resources, you hand in your form to the bureaucrat at that desk, and he gives you another form. This one says, ‘Take this form to Information Management or Marketing.’ Hmm, there is now a choice. Let’s say you decide to go to Information Management. You hand in your form and receive one that says, ‘Take this form to Employee Benefits or Marketing.’ You decide on Employee Benefits where you receive a form saying, ‘Take this form to Corporate Compliance or Human Resources.’

Of the 30 participants, half gave up fairly soon, but the rest kept going until they’d solved it, taking 45 minutes on average. Here’s an online version with four desks, and here’s a fuller description of the project and its variants, including a Kafkaesque 2005 version by Wei-Hwa Huang in which the participants don’t know they’re in a maze.

More of Abbott’s logic mazes.


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Podcast Episode 274: Death in a Nutshell
Image: Flickr

In the 1940s, Frances Glessner Lee brought new rigor to crime scene analysis with a curiously quaint tool: She designed 20 miniature scenes of puzzling deaths and challenged her students to investigate them analytically. In this week’s episode of the Futility Closet podcast we’ll describe the Nutshell Studies of Unexplained Death and their importance to modern investigations.

We’ll also appreciate an overlooked sled dog and puzzle over a shrunken state.

See full show notes …

What Needs More Words?

Ecologists often have to estimate the number of unseen species in an ecosystem: If I count x species of butterfly during my time on an island, how many species probably live there that I did not see? In 1975, Stanford statisticians Bradley Efron and Ronald Thisted applied the same question to the works of William Shakespeare: If we take the Bard’s existing works as a sample, what can we infer about the size of his total vocabulary?

Shakespeare’s known works comprise 884,647 words, which fall into 31,534 “types,” or distinguishable arrangements of letters. Efron and Thisted applied two approaches and found that they produced the same estimate: If a new cache of the playwright’s works were discovered today, equal in size to the old, it would likely contain about 11,460 new word types, with an expected error of less than 150.

So how many word types altogether did Shakespeare know? No upper bound is possible, but they established a lower bound of 35,000 beyond the 31,534 already used — in other words, to write the works that we know of, he likely used less than half his total vocabulary.

(Bradley Efron and Ronald Thisted, “Estimating the Number of Unseen Species: How Many Words Did Shakespeare Know?”, Biometrika 63:3 [1976], 435-447.) (Thanks, Brent.)