Time Tables

polish-american system

Elizabeth Peabody hated rote learning. So the 19th-century American educator adopted the “Polish system,” a graphical means to help students recall reams of historical facts. Invented in the 1820s by Antoni Jażwiński and popularized by Józef Bem, the system relies on a series of 10×10 grids. The location of an event gives its date, the symbol that records it shows its type, and its color indicates the nations involved. A student who wanted to indicate that a revolution took place in America in 1776 would choose the grid for the 18th century, find the square for the 76th year (row 7, column 6), and paint a square using the color designating the British colonies in North America. The square indicates insubordination; a triangle would mean a revolt, an X a conspiracy.

The system is largely forgotten today, but it was immensely popular in Europe and North America in the early 18th century. In the 1830s it was approved for use throughout the French educational system, and Peabody toured the United States offering a book of her own, blank charts and a special set of paints. “Instructors who are themselves not well educated in history, may yet dare to undertake to teach chronology with the help of this manual,” she wrote, “and it must be obvious that highly-accomplished teachers can unfold and develop the subject to an indefinite extent.”

“The results of Peabody’s appropriation of the Polish System are both handsome and surprising,” write Daniel Rosenberg and Anthony Grafton in Cartographies of Time (2012). “Surviving copies of the charts in libraries look nothing like one another. Each bears the imprint of an individual student’s imagination.”

“English as She Is Pronounced”

The wind was rough
And cold and blough,
She kept her hands within her mough.

It chill’d her through,
Her nose grough blough
And still the squall the faster flough.

And yet although
There was nough snough,
The weather was a cruel fough.

It made her cough —
Pray do not scough! —
She coughed until her hat blew ough.

Ah, you may laugh,
You silly caugh!
I’d like to beat you with my staugh.

Her hat she caught,
And saught and faught
To put it on and tie it taught.

Try as she might
To fix it tight
Again it flew off like a kight,

Away up high
Into the skigh.
The poor girl sat her down to crigh.

She cried till eight
P.M., so leight!
Then home she went at a greight reight.

— J.H. Walton

A New Dawn


In July 1054 Chinese astronomers saw a reddish-white star appear in the eastern sky, its “rays stemming in all directions.” Yang Weide wrote:

I humbly observe that a guest star has appeared; above the star there is a feeble yellow glimmer. If one examines the divination regarding the Emperor, the interpretation is the following: The fact that the star has not overrun Bi and that its brightness must represent a person of great value. I demand that the Office of Historiography is informed of this.

It’s now believed they were witnessing SN 1054 — the supernova that gave birth to the Crab Nebula.

No Sale

If Chicken McNuggets come in packs of 6, 9, and 20, what’s the largest number of McNuggets that you can’t buy?

Steve Omohundro and Peter Blicher posed this question in MIT Technology Review in May 2002, and Ken Rosato contributed a neat solution.

The answer is 43. To start, notice that we can use the 6-packs and 9-packs to piece together any multiple of 3 other than 3 itself. 43 itself is not divisible by 3, so 6-packs and 9-packs alone won’t get us there, and adding some 20-packs won’t help, since we’d have to add them to a quantity of either 23 or 3, neither of which can be assembled from packs of other sizes. So that shows that 43 itself can’t be reached.

But we still need to show that every larger number can be. Well, we can create all the larger even numbers by adding some quantity of 6-packs to either 36, 38, or 40, and each of those foundations can be assembled from the packs we have (36 = 9 + 9 + 9 + 9, 38 = 20 + 9 + 9, and 40 = 20 + 20). So that takes care of the even numbers. And adding 9 to any of these even numbers will give us any desired odd number above 43, starting with 36 + 9 = 45.

So 43 is the largest number of Chicken McNuggets that can’t be formed by combining 6-packs, 9-packs, and 20-packs.

(I think Henri Picciotto was the first to broach this arresting question, in Games magazine in 1987. Since then, McNuggets have found their way into Happy Meals in 4-piece servings, reducing the largest non–McNugget number to 11. In some countries, though, the 9-piece allotment has been increased to 10 — and in that case there is no largest such number, as no odd quantity can ever be assembled.)



In their 1981 book Facts and Fallacies, Chris Morgan and David Langford note that the biblical reference to the “four corners of the earth” would apply equally well if the world were a tetrahedron.

In a similar spirit, as American airman Joseph Portney was flying over the North Pole in 1968 he wondered, “What if the Earth were … ?” He made sketches of 12 fanciful alternate Earths and gave them to Litton’s Guidance & Control Systems graphic arts group, which created models that were featured in the company’s Pilots and Navigators Calendar of 1969. This made an international sensation, and Portney’s creations were subsequently published for use in classrooms worldwide, inviting students to ponder what life would be like on a cone or a dodecahedron.

Portney graduated from the U.S. Naval Academy and went on to work for Litton on high-altitude navigation problems — for example, designing control systems that could guide an aircraft around one of these strange worlds.

The Internet Archive has the whole complement.


Image: Wikimedia Commons

Suppose I think, after doing my accounts, that I have a large balance at the bank. And suppose you want to find out whether this belief of mine is ‘wishful thinking.’ You can never come to any conclusion by examining my psychological condition. Your only chance of finding out is to sit down and work through the sum yourself. … It is the same with all thinking and all systems of thought. If you try to find out which are tainted by speculating about the wishes of the thinkers, you are merely making a fool of yourself. You must first find out on purely logical grounds which of them do, in fact, break down as arguments. Afterwards, if you like, go on and discover the psychological causes of the error.

You must show that a man is wrong before you start explaining why he is wrong. The modern method is to assume without discussion that he is wrong and then distract his attention from this (the only real issue) by busily explaining how he became so silly. In the course of the last fifteen years I have found this vice so common that I have had to invent a name for it. I call it ‘Bulverism’. Some day I am going to write the biography of its imaginary inventor, Ezekiel Bulver, whose destiny was determined at the age of five when he heard his mother say to his father — who had been maintaining that two sides of a triangle were together greater than a third — ‘Oh you say that because you are a man.’ ‘At that moment’, E. Bulver assures us, ‘there flashed across my opening mind the great truth that refutation is no necessary part of argument. Assume that your opponent is wrong, and explain his error, and the world will be at your feet. Attempt to prove that he is wrong or (worse still) try to find out whether he is wrong or right, and the national dynamism of our age will thrust you to the wall.’ That is how Bulver became one of the makers of the Twentieth Century.

— C.S. Lewis, “Bulverism: or, The Foundation of Twentieth-Century Thought,” 1941


Consider the set (2, 5, 9, 13). Which of these numbers can be tossed out, and for what reason?

We might choose:

  • 2 because it’s the only even number.
  • 9 because it’s the only non-prime.
  • 13 because it doesn’t fit in the sequence AnAn-1 = 1 + (An-1An-2).

“Hence one could toss out either 2, 9 or 13,” observes Marquette University mathematician George R. Sell. “Therefore one should toss out 5 because it is the only number that cannot be tossed out.”

(George R. Sell, “A Paradox,” Pi Mu Epsilon Journal 2:6 [Spring 1957], 278.)

“A Matter of Opinion”


A man walks round a pole on the top of which is a monkey. As the man moves, the monkey turns on the top of the pole, so as still to keep face to face with the man. Now, when the man has gone round the pole, has he or has he not gone round the monkey?

— John Scott, The Puzzle King, 1899

A Path-Making Game

Alice and Bob are playing a game. An n×n checkerboard lies between them. Alice begins by marking a corner square, and thereafter the two of them take turns marking squares; each one they choose must be adjacent orthogonally to the last one chosen, so together they’re making a path around the board. When the path can’t continue (because no unmarked adjacent square is available), then the player who moved last wins. For which n can Alice devise a winning strategy? What if she has to start by marking a square adjacent to a corner, rather than the corner itself?

Click for Answer


A Highwayman confronted a Traveler, and covering him with a firearm, shouted: ‘Your money or your life!’

‘My good friend,’ said the Traveler, ‘according to the terms of your demand my money will save my life, my life my money; you imply that you will take one or the other, but not both. If that is what you mean please be good enough to take my life.’

‘That is not what I mean,’ said the Highwayman; ‘you cannot save your money by giving up your life.’

‘Then take it anyhow,’ the Traveler said. ‘If it will not save my money it is good for nothing.’

The Highwayman was so pleased with the Traveler’s philosophy and wit that he took him into partnership and this splendid combination of talent started a newspaper.

— Ambrose Bierce, Fantastic Fables, 1899