The Ulam Spiral

Write the numbers from 41 to 440 in a square spiral:

ulam spiral

Remarkably, all the numbers on the red diagonal are prime — even when the spiral is continued into a 20 × 20 square.

No one’s quite sure what to make of this. Polish mathematician Stanislaw Ulam discovered the pattern while doodling at a scientific meeting in 1963.

The Self-Flowing Flask'sSelfFlowingFlask.png

Denis Papin described this perpetual motion scheme in 1685. The pint of liquid in the goblet weighs more than the ounce in the tube, and so forces it down. When the liquid spills out the top of the tube and back into the goblet, it starts a cycle that must continue until the liquid finally evaporates. Where is the error?

Pieces of Pi

In the decimal expansion of π:

  • the digits 27182818 — the first eight digits of e — appear at position 1,526,800.
  • the digits 14142135 — the first eight digits of the square root of 2 — appear at position 52,638.
  • the first eight digits of π itself — 31415926 — reappear at position 50,366,472.
  • 16470 appears at position 16470.
  • there are seven 7s at position 3,346,228, eight 8s at position 46,663,520, and six 9s at position 762.
  • The White House switchboard number (456-1414) is at position 3,193,808, the population of France (65,073,482) is at position 98,709,092, and Disneyland’s zip code (92802) is at position 41,112.

Write out the alphabet starting with J:


Erase all letters that have left-right symmetry (such as A) and count the letters in each of the five groups that remain. (James Davis)

The Bystander Paradox
Image: Wikimedia Commons

You and I are walking through a dark alley. You are explaining that it’s wrong to kill an innocent person, but that it’s morally permissible to kill someone who is threatening your life.

At that moment an attacker leaps out of the shadows — and you find that, because of my position, you cannot defend yourself without also killing me.

In that instant, what is your moral obligation? You must not kill an innocent man. But my presence is threatening your life, so in that sense I am not innocent and can be sacrificed.

But if there’s no imperative to spare my life, then I’m no longer a hindrance to your self-defense … in which case I’m innocent again and you’re not justified in killing me.

And so on.

Euclid, Schmeuclid

euclid schmeuclid fallacy

Consider any two circles intersecting at points Q and R. Their diameters are QP and QS. PS intersects the circles at M and N.

Now, any angle inscribed in a semicircle is a right angle. This means that both QNP and QMS are right angles — and thus that there are two perpendiculars from Q to PS. Doesn’t it?

Mrs. Miniver’s Problem

mrs. miniver's problem

One of Jan Struther’s popular stories of the 1930s included the following passage:

She saw every relationship as a pair of intersecting circles. It would seem at first glance that the more they overlapped the better the relationship; but this is not so. Beyond a certain point the law of diminishing returns sets in, and there are not enough private resources left on either side to enrich the life that is shared. Probably perfection is reached when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle. On paper there must be some neat mathematical formula for arriving at this; in life, none.

Interestingly, mathematicians who pursued the problem found that no precise solution is possible. With circles, as with relationships, we have to do the best we can.

Hard Bargaining

In Robert Louis Stevenson’s story “The Bottle Imp,” the titular imp will grant its owner (almost) any wish, but if the owner dies with the bottle then he burns in hell. He may sell the bottle, but he must charge less than he paid for it, and the new buyer must understand these conditions.

Now, no one would buy such a bottle for 1 cent, as he could not then sell it again. (The imp can’t make you immortal, or support prices smaller than one cent, or alter the conditions.) And if 1 cent is too low a price, then so is 2 cents, for the same reason. And so on, apparently forever. It would be irrational to buy the bottle for any price.

But intuitively most people would consider $1,000 a reasonable price to pay for the use of a wish-granting genie. Who’s right?

See also Tug of War.

The Wealth Puzzle

A beggar asks a wealthy man for a penny. He says he wants to be a wealthy man himself someday.

“Well, consider,” says the wealthy man. “A single penny will not make you wealthy. And a second penny will not make you wealthy either.

“Indeed, there is no point at which an additional penny will make you a wealthy man. So I’m sorry, my friend, but your dream is impossible.”

Poetry in Motion"An+inextensible+heavy+chain"+maxwell&as_brr=1&ei=iQh6SoqCMZ6SygSHkbzKDA#v=onepage&q=&f=false

When he wasn’t taming electromagnetism, James Clerk Maxwell wrote verse. Here’s “A Problem in Dynamics,” composed in 1854:

An inextensible heavy chain
Lies on a smooth horizontal plane,
An impulsive force is applied at A,
Required the initial motion of K.

Let ds be the infinitesimal link,
Of which for the present we’ve only to think;
Let T be the tension, and T + dT
The same for the end that is nearest to B.
Let a be put, by a common convention,
For the angle at M ‘twixt OX and the tension;
Let Vt and Vn be ds‘s velocities,
Of which Vt along and Vn across it is;
Then Vn/Vt the tangent will equal,
Of the angle of starting worked out in the sequel.

In working the problem the first thing of course is
To equate the impressed and effectual forces.
K is tugged by two tensions, whose difference dT
Must equal the element’s mass into Vt.
Vn must be due to the force perpendicular
To ds‘s direction, which shows the particular
Advantage of using da to serve at your
Pleasure to estimate ds‘s curvature.
For Vn into mass of a unit of chain
Must equal the curvature into the strain.

Thus managing cause and effect to discriminate,
The student must fruitlessly try to eliminate,
And painfully learn, that in order to do it, he
Must find the Equation of Continuity.
The reason is this, that the tough little element,
Which the force of impulsion to beat to a jelly meant,
Was endowed with a property incomprehensible,
And was “given,” in the language of Shop, “inextensible.”
It therefore with such pertinacity odd defied
The force which the length of the chain should have modified,
That its stubborn example may possibly yet recall
These overgrown rhymes to their prosody metrical.
The condition is got by resolving again,
According to axes assumed in the plane.
If then you reduce to the tangent and normal,
You will find the equation more neat tho’ less formal.
The condition thus found after these preparations,
When duly combined with the former equations,
Will give you another, in which differentials
(When the chain forms a circle), become in essentials
No harder than those that we easily solve
In the time a T totum would take to revolve.

Now joyfully leaving ds to itself, a-
Ttend to the values of T and of a.
The chain undergoes a distorting convulsion,
Produced first at A by the force of impulsion.
In magnitude R, in direction tangential,
Equating this R to the form exponential,
Obtained for the tension when a is zero,
It will measure the tug, such a tug as the “hero
Plume-waving” experienced, tied to the chariot.
But when dragged by the heels his grim head could not carry aught,
So give a its due at the end of the chain,
And the tension ought there to be zero again.
From these two conditions we get three equations,
Which serve to determine the proper relations
Between the first impulse and each coefficient
In the form for the tension, and this is sufficient
To work out the problem, and then, if you choose,
You may turn it and twist it the Dons to amuse.