Traffic Planning

traffic planning - cars

Towns A and B are connected by two roads. Suppose that two cars connected by a rope of length 2r can travel from A to B without breaking the rope. How can we prove that two circular wagons of radius r, moving along these roads in opposite directions, will necessarily collide?

traffic planning - wagons

This can be solved neatly by creating a configuration space. Map each road onto a unit segment, and set these up as two sides of a square. The northern car’s progress is reflected by a point moving up the left side of the square, and the southern car’s by a point moving from left to right along the bottom. Now the motion of the two cars from A to B is represented by a continuous curve connecting (0,0) and (1,1).

The wagons start from opposite towns, so their motion is represented by a curve from (0,1) to (1,0), and it’s immediately clear that the two curves must intersect. The intersection point corresponds to the collision of the wagons.

This example, by N. Konstantinov, is reportedly common in Russian mathematical folklore; I found it in Serge Tabachnikov’s 2005 book Geometry and Billiards (of all places).

Image: Wikimedia Commons

SIR, — While surveying in northern Labrador I had occasion to visit the island of Nukusustok, a few miles to seaward of the village of Nain. On the slope of a hill, and about 300ft. inland, I found a golf ball in good condition. How did the ball come to be there, and so far inland? It is possible that the ball was driven by a golfer from an Atlantic liner during practice, drifted northward past Greenland, and was finally carried ashore by the Labrador current which runs from north to south along the Labrador coast.

I have sent the ball to Dunlops, the makers, who suggest that it was probably carried so far inland by a sea bird. Perhaps some of your readers could help in explaining the mystery.

Yours faithfully,

Thos. O. Hampson

The Field, June 29, 1935

A Compensatory Harmonica
Image: Flickr

A problem from the American Mathematical Monthly, March 1930:

Two men jointly own x cows. They sell these for x dollars per head and use the proceeds to buy some sheep at $12 per head. Their income from the cows isn’t divisible by 12, so they buy a lamb with the remainder. Later they divide the flock so that each man has the same number of animals. This leaves the man with the lamb somewhat short-changed, so the other man gives him a harmonica. What’s the harmonica worth?

Click for Answer

The Spring (Arrangements) Bill

In 1936 English humorist A.P. Herbert found himself sitting in Parliament as an independent member for Oxford University. He drafted the following bill in verse to honor the new season — it’s a shame that it wasn’t enacted:

Whereas in every lawn and bed the plucky crocus lifts his head, and to and fro sweet song-birds go, the names of which we do not know:

Whereas the woods no more are dumb, the Boat Race and the Budget come, the Briton swells his manly chest, his mate, as eager, scrubs the nest, and Spring, with light but lavish hand, is spreading madness o’er the land:

It is expedient — but in rhyme — to legislate for such a time: Be it enacted, therefore, by our King with Lords and Commons in a fairy ring, assembled joyously at Westminister (or any other place that they prefer):

Provision for a Season Called Spring

1. (i) It shall be lawful everywhere for citizens to walk on air, to hang their hats upon the trees and wander hatless if they please: and notwithstanding any cracked provision in a previous Act, to give a constable a kiss is not felonious after this.

(ii) All citizens who choose to ride on taxi-tops and not inside: and those who do not use their votes because they’re busy painting boats: and any miscreant who hums, instead of doing dismal sums: whoever does a silly thing need only answer “‘Tis the Spring”: and this shall be a good defence in any court with any sense:

Provided that, in late July, this Act, of course, does not apply.

Financial Provisions

2. If any person feels he must get out of London now or bust, because the Spring is in his bones, but he must work for Mr. Jones, it shall be lawful for the same to give the Treasury his name, and say “Upon sufficient grounds I want about a hundred pounds”: and there shall not be any fuss concerning sums expended thus.

Repeal of Redundant Statutes

3. Subsection (i) of Section Four of any Act that seems a bore, and all the Acts concerning beer, and every Act that is not clear (always excepting Schedule A), shall be repealed and thrown away.

House of Commons — Reform of Procedure — Music etc.

4. (i) There shall be banks of maidenhair arranged about the Speaker’s chair: and roses white and roses red shall hang above the Speaker’s head: like some tremendous window-box, the Galleries be gay with phlox: and goldfish, lovely but aloof, shall swim above the glassy roof.

(ii) From now until the First of June all speeches shall be sung (in tune). The Speaker shall determine what hon. Members are in tune or not.

(iii) When in Committee of Supply the House may hum (but not too high). The Clerk-Assistant-at-the-Table shall choose the key (if he is able).

(iv) A band shall nearly always play (not on the first Allotted Day) behind the Speaker’s Chair at three and on the Terrace after tea.

Saving for Committees

5. On any day in May or June Committees shall adjourn quite soon: Provided, if the cuckoo call, Committees shall not sit at all.

Sittings of the Upper House

6. The House of Lords shall never sit on sunny days till after Whit: and they shall rise, if they have met, when it is foggy, fine or wet.

Termination of Official Report

7. (i) Except as hereinafter hinted, Hansard shall not again be printed, and save as in this Act is learned, all previous Hansards shall be burned.

(ii) It is a pity, history teaches, to make reports of people’s speeches, and afterwards to be unkind, simply because they change their mind. It is a most disgusting thing to make such comments in the Spring: so, as from when this Act is passed, that day’s Report shall be the last.

(iii) And as regards exceptions, see Subheading (a) of Schedule B.

Powers and Duties of Departments

8. (i) The secretary of State for Home Affairs shall now proceed to Rome, to Moscow, Washington, Cathay, or anywhere that’s far away, and not return to English skies until the Speaker certifies that Spring has ceased to be a fact under the Moss (Collection) Act.

(ii) Meanwhile o’er all his grim domain a lovely golden girl shall reign: and this delicious creature shall give golden parties in the Mall (paying the bills, if she is dunned, from the Consolidated Fund). The Civil Service, hand in hand, shall dance in masses down the Strand: and all the Cabinet shall wear wild dandelions in their hair.

(iii) It shall be deemed that every one has come into the world for fun. This shall be printed on the wall of every office in Whitehall.

Penalties for Certain Expressions

9. (i) No kind of crisis shall excuse a man exploring avenues: no lesser doom does he deserve when he is straining every nerve: and special punishment is earned by those who leave no stone unturned.

(ii) The penalty for each offence shall be elastic but immense.

(iii) A pension shall reward the man who modestly does all he can.


10. (i) The greatest care has been employed to make this measure null and void: not one expression in this Act means anything it means in fact.

(ii) Examples we decline to give: the lawyers, after all, must live.


11. This Act applies and shall be good where anybody thinks it should:

Provided that, if strong objection should be expressed to any Section, that Section shall not have effect except for those who don’t object.


Any speech, motion, question, amendment or interruption by


The Senster

In September 1970, cybernetic sculptor Edward Ihnatowicz unveiled a remarkable piece of robotic art at a Dutch science museum. Standing 8 feet high at the shoulder and “resembling a giraffe or dinosaur,” the Senster was basically a mechanical lobster claw mounted on a six-jointed neck actuated by quiet hydraulic rams. Using an array of microphones, the creature would turn its head in the direction of a sound, its speed proportional to the volume. If the direction of the sound source remained constant, the rest of the body would gradually follow, making the “animal” appear to home in on the sound. It would shy away from loud noises, and at overwhelming sound levels it would raise its neck vertically and “disdainfully” ignore further sounds until the volume came down. Doppler radar units enabled it to detect the motion of visitors; it was attracted toward small motions but “frightened of sudden movements.” All of this ran on 8K of core memory, but Ihnatowicz found that visitors quickly imputed an animal-like intelligence to the sculpture, and the atmosphere of the exhibit was much like that at a zoo:

In the quiet of the early morning the machine would be found with its head down, listening to the faint noise of its own hydraulic pumps. Then, if a girl walked by, the head would follow her, looking at her legs. Ihnatowicz describes his own first stomach-turning experience of the machine when he had just got it working: he unconsciously cleared his throat, and the head came right up to him as if to ask, ‘Are you all right?’ He also noticed a curious aspect of the effect the Senster had on people. When he was testing it he gave it various random patterns of motion to go through. Children who saw it operating in this mode found it very frightening, but no one was ever frightened when it was working in the museum with its proper software, responding to sounds and movement.

MIT roboticist Rodney Brooks later suggested that intelligent behavior can be achieved when sensory signals are mapped as directly as possible to motor signals through a large number of loosely coupled processes, with minimal internal processing. The Senster wasn’t updating an internal model of the world; it would simply turn its head toward a sound, but its behavior struck visitors as intelligent.

(Aleksandar Zivanovic, “The Technologies of Edward Ihnatowicz,” in Paul Brown et al., eds., White Heat Cold Logic: British Computer Art 1960-1980, 2008.)


Composition 1960 #5, by avant-garde composer La Monte Young:

Turn a butterfly (or any number of butterflies) loose in the performance area.

When the composition is over, be sure to allow the butterfly to fly away outside.

The composition may be any length, but if an unlimited amount of time is available, the doors and windows may be opened before the butterfly is turned loose and the composition may be considered finished when the butterfly flies away.

“I felt certain the butterfly made sounds,” Young wrote, “not only with the motion of its wings but also with the functioning of its body … and unless one was going to dictate how loud or soft the sounds had to be before they could be allowed into the realm of music … the butterfly piece was music.”

In Visible Deeds of Music (2002), Simon Shaw-Miller writes, “An insect recognized as of great beauty, often understood as a symbol of transformation in art, is here the instrument itself. Its flight acts as a visual metaphor for the absent melody, or inaudible sound; Young is reported to have said to his colleague Tony Conrad, ‘Isn’t it wonderful if someone listens to something he is ordinarily supposed to look at?'”

The Humble-Nishiyama Randomness Game

Mathematicians Steve Humble and Yutaka Nishiyama invented this game to highlight a surprising result in probability, based on a principle discovered by Walter Penney.

Two players play the game using an ordinary deck of cards. The cards will be dealt out in a row, one after another. Before the dealing begins, each player claims an ordered sequence of colors that might turn up, for example “red black red” (RBR) or “black red red” (BRR). As the cards are dealt, if three successive cards turn up in one of these sequences, the player who claimed it gets to collect those three cards as a trick, and the dealing continues. When all 52 cards have been dealt, the player who has collected the most tricks wins. (In a typical game, 7 to 9 tricks are won.)

This sounds like a perfectly even game, but in fact the second player has a strategy that will given him a significant advantage. When the first player has chosen his sequence (say, RRB), the second player changes the middle color, adds it to the start of the sequence, discards the last color, and claims the resulting sequence (in this case, BRR). This gives a decided advantage to the second player no matter which sequence his opponent has chosen. In a computer simulation of 1,000 games, Humble and Nishiyama got these results:

BBB vs RBB – RBB wins 995 times, 4 draws, BBB wins once
BBR vs RBB – RBB wins 930 times, 40 draws, BBR wins 30 times
BRB vs BBR – BBR wins 805 times, 79 draws, RBR wins 116 times
RBB vs RRB – RRB wins 890 times, 68 draws, RBB wins 42 times
BRR vs BBR – BBR wins 872 times, 65 draws, BRR wins 63 times
RBR vs RRB – RRB wins 792 times, 85 draws, RBR wins 123 times
RRB vs BRR – BRR wins 922 times, 51 draws, RRB wins 27 times
RRR vs BRR – BRR wins 988 times, 6 draws, RRR wins 6 times

Penney’s original game uses coin flips; cards are preferable because no record-keeping is required and because the finite number of cards in a deck increases the second player’s chances.

(Steve Humble and Yutaka Nishiyama, “Humble-Nishiyama Randomness Game — A New Variation on Penney’s Coin Game,” Mathematics Today, August 2010.)