You own a goat and a meadow. The meadow is in the shape of an equilateral triangle each side of which is 100 meters long. The goat is tied to a post at one corner of the meadow. How long should you make the tether in order to give the goat access to exactly half the meadow?
Ellen Terry played Juliet at London’s Lyceum Theatre in 1882. The following was later found on the flyleaf of her copy of the text:
Get the words into your remembrance first of all. Then, (as you have to convey the meaning of the words to some who have ears, but don’t hear, and eyes, but don’t see) put the words into the simplest vernacular. Then exercise your judgment about their sound.
So many different ways of speaking words! Beware of sound and fury signifying nothing. Voice unaccompanied by imagination, dreadful. Pomposity, rotundity.
Imagination and intelligence absolutely necessary to realize and portray high and low imaginings. Voice, yes, but not mere voice production. You must have a sensitive ear, and a sensitive judgment of the effect on your audience. But all the time you must be trying to please yourself.
Get yourself into tune. Then you can let fly your imagination, and the words will seem to be supplied by yourself. Shakespeare supplied by oneself! Oh!
Realism? Yes, if we mean by that real feeling, real sympathy. But people seem to mean by it only the realism of low-down things.
To act, you must make the thing written your own. You must steal the words, steal the thought, and convey the stolen treasure to others with great art.
(From Donald Sinden, ed., The Everyman Book of Theatrical Anecdotes, 1987.)
Based on an ancient Hindu game, Snakes and Ladders (Chutes and Ladders in ophidiophobic America) is at heart a morality lesson: As you progress by die roll from square 1 to square 100 and spiritual enlightenment, your way is complicated by virtues and vices. Landing on a snake (or chute) will send you back to an earlier square, and landing on a ladder will send you ahead to a later one. Each of these shortcuts is associated with a precept — “Carelessness” leads to “Injury,” “Study” leads to “Knowledge,” and so on.
In 1993 University of Michigan mathematician S.C. Althoen and his colleagues considered the game as a 101-state absorbing Markov chain. The shortest possible game lasts seven moves, the longest is infinite, and according to their calculations the expected number of moves in the Milton Bradley version of Chutes and Ladders is
which is about 39.2.
Troublingly, the average length of a game without snakes or ladders (just the 100-square board) is almost exactly 33 moves: “Apparently the snakes lengthen the game more than the ladders shorten it.” And, while adding a ladder will generally shorten the game and adding a snake will lengthen it, this isn’t always the case: In the original game, adding a ladder from square 79 to square 81 lengthens the expected playing time by more than two moves (to about 41.9), since it increases the chance of missing the important ladder leading from square 80 to square 100. And adding a snake from square 29 to square 27 shortens the game by more than a move (to about 38.0), since it offers a second chance at the long ladder from 28 to 84.
So, arguably, we might advance more quickly through life with more vice and less virtue.
(S.C. Althoen, L. King, and K. Schilling, “How Long Is a Game of Snakes and Ladders?”, Mathematical Gazette 77:478 [March 1993], 71-76.)
If White withdraws his light-squared bishop along the b1-h7 diagonal, then Black is facing two mate threats: White can move his king off the a-file, discovering mate by the a8 rook, or he can move his dark-squared bishop off the first rank, discovering mate by the h1 queen. A few sample lines:
1. Bf5 Rxa8+ 2. Ba7#
1. Bf5 Rxh1 2. Kb3#
1. Bf5 Rh4 [ready to block a rook check or take the queen] 2. Bh2#
1. Bf5 Rd8 [ready to block a queen check or take the rook] Bd4#
(If 1. Bf5 c2 then 1. Bd4#!)
At first it looks as though Black is doomed, but he does have one resource. If after e.g. 1. Bf5 he plays 1. … Rg8 then White has no way to mate on the next move: If the white king discovers check, Black can simply take the rook, and if the g1 bishop discovers check by the queen, the rook can interpose.
The answer is 1. Bg6!, the only initial bishop move that blocks the rook’s route to g1 and keeps both mate threats alive.
On its 35th anniversary, THX, the sound quality assurance company founded by George Lucas, released the original score of “Deep Note,” its audio trademark, which debuted at the premiere of Return of the Jedi in 1983 and is now familiar from countless films. Essentially it’s a stupendous D chord; the U.S. trademark registration reads:
The THX logo theme consists of 30 voices over seven measures, starting in a narrow range, 200 to 400 Hz, and slowly diverting to preselected pitches encompassing three octaves. The 30 voices begin at pitches between 200 Hz and 400 Hz and arrive at pre-selected pitches spanning three octaves by the fourth measure. The highest pitch is slightly detuned while there are double the number of voices of the lowest two pitches.
“I like to say that the THX sound is the most widely-recognized piece of computer-generated music in the world,” says James A. Moorer, who wrote it. “This may or may not be true, but it sounds cool.” And now that we have the score you can do this:
Created by Franz Armbruster in 1967, “Instant Insanity” was the Rubik’s Cube of its day, a simple configuration task with a dismaying number of combinations. You’re given four cubes whose faces are colored red, blue, green, and yellow:
The task is to arrange them into a stack so that each of the four colors appears on each side of the stack. This is difficult to achieve by trial and error, as the cubes can be arranged in 41,472 ways, and only 8 of these give a valid solution.
One approach is to use graph theory — draw points of the four face colors and connect them to show which pairs of colors fall on opposite faces of each cube:
Then, using certain criteria (explained here), we can derive two directed subgraphs that describe the solution:
The first graph shows which colors appear on the front and back of each cube, the second which colors appear on the left and right. Each arrow represents one of the four cubes and the position of each of the two colors it indicates. So, for example, the black arrow at the top of the first graph indicates that the first cube will have yellow on the front face and blue on the rear.
This solution isn’t unique, of course — once you’ve compiled a winning stack you can rotate it or rearrange the order of the cubes without affecting its validity. B.L. Schwartz gives an alternative method, through inspection of a table, as well as tips for solving by trial and error using physical cubes, in “An Improved Solution to ‘Instant Insanity,'” Mathematics Magazine 43:1 (January 1970), 20-23.