## Stacking Cubes

4^{3} + 18^{3} + 33^{3} = 41833

16^{3} + 50^{3} + 33^{3} = 165033

22^{3} + 18^{3} + 59^{3} = 221859

44^{3} + 46^{3} + 64^{3} = 444664

48^{3} + 72^{3} + 15^{3} = 487215

98^{3} + 28^{3} + 27^{3} = 982827

## Dark Doings

There is a legend that after Buddha died, his shadow lingered in a cave. It actually is possible for a shadow to persist without any sustaining object. Light travels at 299,792,458 meters per second in a vacuum. The moon is about 384,400,000 meters away from Earth. Hence, if the moon were instantly obliterated during a solar eclipse, its shadow would linger for more than a second on the surface of Earth. If the moon were farther away, its shadow could last several minutes. We can extrapolate to posthumous shadows that postdate their objects by millions of years. We can also speculate about an infinite past in which a shadow is sustained by a beginningless sequence of objects. As one object is destroyed, an object of the same shape and size seamlessly replaces it. This shadow antedates any object in the sequence and so refutes the principle that every shadow is caused by an object. Shadows are not dedicated dependents. Although slaves to some object or other, they can switch masters.

– Roy Sorensen, *Seeing Dark Things*, 2008

## Franklin the Wizard

But the sage was not too grave to play a joke on his friends. One day, when they were walking in the park at Wycombe, he said that he could quiet the waves on a small stream which was being whipped by the wind. He went two hundred paces above where the others stood, made some magic passes over the water, and waved his bamboo cane three times in the air. The waves gradually sank and the stream became as smooth as a mirror. After they had marvelled Franklin explained. He carried oil in the hollow joint of his cane, and a few drops of it spreading on the water had caused the miracle.

– Carl Van Doren, *Benjamin Franklin*, 1938

## Chasing Leo

In 1938, the *American Mathematical Monthly* published an unlikely paper: “A Contribution to the Mathematical Theory of Big Game Hunting.” In it, Ralph Boas and Frank Smithies presented 16 ways to catch a lion using techniques inspired by modern math and physics. Examples:

- “The Method of Inversive Geometry. We place a
*spherical*cage in the desert, enter it, and lock it. We perform an inversion with respect to the cage. The lion is then in the interior of the cage, and we are outside.” - “A Topological Method. We observe that a lion has at least the connectivity of the torus. We transport the desert into four-space. It is then possible to carry out such a deformation that the lion can be returned to three-space in a knotted condition. He is then helpless.”
- “The Dirac Method. We observe that wild lions are,
*ipso facto*, not observable in the Sahara Desert. Consequently, if there are any lions in the Sahara, they are tame. The capture of a tame lion may be left as an exercise for the reader.” - “A Relativistic Method. We distribute about the desert lion bait containing large portions of the Companion of Sirius. When enough bait has been taken, we project a beam of light across the desert. This will bend right round the lion, who will then become so dizzy that he can be approached with impunity.”

The article has inspired a tradition of updates by other mathematicians over the years:

- “Let
*Q*be the operator that encloses a word in quotation marks. Its square*Q*^{2}encloses a word in double quotes. The operator clearly satisfies the law of indices,*Q*^{m}*Q*^{n}=*Q*^{m + n}. Write down the word ‘lion,’ without quotation marks. Apply to it the operator*Q*^{-1}. Then a lion will appear on the page. It is advisable to enclose the page in a cage before applying the operator.” (I.J. Good, 1965) - “Game Theoretic Method. A lion is big game. Thus,
*a fortiori*, he is a game. Therefore there exists an optimal strategy. Follow it.” (“Otto Morphy,” 1968) - “Method of Analytics Mechanics. Since the lion has nonzero mass it has moments of inertia. Grab it during one of them.” (Patricia Dudley et al., 1968)
- “Method of Natural Functions. The lion, having spent his life under the Sahara sun, will surely have a tan. Induce him to lie on his back; he can then, by virtue of his reciprocal tan, be cot.” (Dudley)
- “Nonstandard Analysis. In a nonstandard universe (namely, the land of Oz), lions are cowardly and may be caught easily. By the transfer principle, this likewise holds in our (standard) universe.” (Houston Euler, et al., 1985)

Dudley also suggested a “method of moral philosophy”: “Construct a corral in the Sahara and wait until autumn. At that time the corral will contain a large number of lions, for it is well known that a pride cometh before the fall.”

## The Guzman Prize

In December 1900, a French committee offered 100,000 francs to the first person to make contact with intelligent beings on another planet.

Martians were excluded as too easy.

(Thanks, Tom.)

## “Another Paradox”

If a cork ball about an inch in diameter be tied at the end of a thread about a foot in length, and then swung so that it enters a smooth stream of water flowing from a tap at about three inches from the mouth of the latter, it will be found that the ball will remain in the water, and that the thread will make an angle of about thirty degrees with a vertical line passing through the ball. The latter, it should be added, must be thoroughly wetted before this result is produced.

– *Strand*, September 1908

## Misc

- ALONE and UPRAISE can be beheaded twice and retain their essential meanings.
- Evelyn Waugh’s first wife was named Evelyn.
- 1033 = 8
^{1}+ 8^{0}+ 8^{3}+ 8^{3} - Can a shadow rotate?
- “Genius is nothing but a greater aptitude for patience.” — Benjamin Franklin

## Math Notes

20864448472975628947226005981267194447042584001 = (2 + 0 + 8 + 6 + 4 + 4 + 4 + 8 + 4 + 7 + 2 + 9 + 7 + 5 + 6 + 2 + 8 + 9 + 4 + 7 + 2 + 2 + 6 + 0 + 0 + 5 + 9 + 8 + 1 + 2 + 6 + 7 + 1 + 9 + 4 + 4 + 4 + 7 + 0 + 4 + 2 + 5 + 8 + 4 + 0 + 0 + 1)^{20}

## Moving Without Motion?

Achilles-weed is prostrate and grows along the ground at the amazing rate of 10 cm per hour. An exceeding slow tortoise munches one end of the Achilles-weed at the same rate as it grows at the other end. So the tortoise appears to chase the Achilles-weed round the garden. But, strictly speaking, the Achilles-weed does not move at all, it grows and is eaten. Yet its location changes, and it is made up of parts whose location changes (the left and right-hand halves of the Achilles-weed). Hence being made up of parts whose location changes is not sufficient for motion.

– Peter Forrest, “Is Motion Change of Location?”, *Analysis*, 1984

## A Pretty Problem

In Longfellow’s novel *Kavanagh*, Mr. Churchill reads a word problem to his wife:

“In a lake the bud of a water-lily was observed, one span above the water, and when moved by the gentle breeze, it sunk in the water at two cubits’ distance. Required the depth of the water.”

“That is charming, but must be very difficult,” she says. “I could not answer it.”

Is it? If a span is 9 inches and a cubit is 18 inches, how deep is the water?

## Misc

- SWEET-TOOTHED has three consecutive pairs of letters. SUBBOOKKEEPER has four.
- Will you answer this question negatively?
- 4624 = 4
^{4}+ 4^{6}+ 4^{2}+ 4^{4} - The telephone number 278-7433 spells both ASTRIDE and CRUSHED.
- “Nothing great was ever achieved without enthusiasm.” — Emerson

## Beach Reading

Amazon reviews of *A Million Random Digits with 100,000 Normal Deviates* (1955), by the RAND Corporation:

- “I had a hard time getting into this book. The profanity was jarring and stilted, not at all how people really talk.”
- “Once you get about halfway in, the rest of the story is pretty predictable.”
- “If you like this book, I highly recommend that you read it in the original binary.”
- “I would have given it five stars, but sadly there were too many distracting typos. For example: 46453 13987.”
- “I really liked the ’10034 56429 234088′ part.”
- “Frankly the sex scenes were awkward and clumsily written, adding very little of value to the plot.”
- “For a supposedly serious reference work the omission of an index is a major impediment. I hope this will be corrected in the next edition.”

The average customer gives it four stars.

## Misc

- EVIAN, SEIKO, and STROH’S are all English words spelled backward.
- Can “I apologize” be false?
- 165033 = 16
^{3}+ 50^{3}+ 33^{3} - Little Wymondley, in Hertfordshire, is bigger than Great Wymondley.
- “How old would you be if you didn’t know how old you was?” — Satchel Paige

## A Surprise Visitor

Create two columns, one starting with the numbers 12 and 18 and the other with 5 and 5. Continue each column, deriving each new number by adding the two that precede it:

In the *Journal of Recreational Mathematics*, James Davis writes, “Forming successive pairs with adjoining numbers from each column one finds the ratio of the two numbers in each pair converges to Pi!” How can this be?

“The alert reader will suspect there is a trick in this method, as I did when Pi first presented it to me. The labor of several hours of computation coupled with trial and error produced half of the secret of the method. It is obviously based somehow on the fact that Phi (the golden mean, which equals (1 + (sqrt 5))/2), can be closely approximated by the nifty pseudo equation below:”

1.2 x Phi^2 = Pi

“Can the reader decipher Pi’s technique for making herself with Phi?”

## Through and Through

3139971973786634711391448651577269485891759419122938744591877656925789747974914319

422889611373939731 is prime, whether it’s spelled forward or backward. Further, if it’s cut into 10 pieces:

… each row, column, and diagonal is itself a reversible prime.

Discovered by Jens Kruse Andersen.

## A Man’s Home …

In 1922, after the death of his mother, Carl Jung felt “I had to achieve a kind of representation in stone of my innermost thoughts and of the knowledge I had acquired. Or, to put it another way, I had to make a confession of faith in stone.”

He began to build a structure on the shores of Lake Zurich in Switzerland. It began as a regular two-story house, “a maternal hearth,” but over the years he added a towerlike annex with a “retiring room” for withdrawal and contemplation, and a courtyard and loggia.

At 80, after his wife’s death, “I suddenly realized that the small central section which crouched so low, so hidden, was myself!” He added an upper story, an extension of his own personality no longer hidden behind the “maternal” and “spiritual” towers. “Now it signified an extension of consciousness achieved in old age. With that the building was complete.”

The final building, he saw, symbolized the structure of his own psyche, the full emergence of his personality in adulthood. “Unconsciously built at the time, only afterward did I see how all the parts fitted together and that a meaningful form had resulted: a symbol of psychic wholeness.” “At Bollingen,” he wrote, “I am in the midst of my true life, I am most deeply myself.”

## Petty Cash

I place $20 in a box.

So do you.

Now the box contains $40, and we both know it.

I sell the box to you for $30.

And we both walk away with a $10 profit.

(Thanks, Ben.)

## Hocus Pocus

Choose any number and write down its divisors:

14

1 2 7 14

Then write down the number of divisors of each of these divisors:

14

1 2 7 14

1 2 2 4

Now the square of the sum of this last group will always equal the sum of its members’ cubes:

(1 + 2 + 2 + 4)^{2} = 1^{3} + 2^{3} + 2^{3} + 4^{3}

Discovered by Joseph Liouville.

## Misc

- The smallest number name that’s typed with eight fingers is ONE SEPTILLION ONE THOUSAND.
- Cincinnati, Cleveland, and Columbus are all towns in Indiana.
- 2427 = 2
^{1}+ 4^{2}+ 2^{3}+ 7^{4} - SAN DIEGO is an anagram of DIAGNOSE.
- “It is not customary to love what one has.” — Anatole France

## Whole Lotta Nothing

Do holes exist? That is, if a hole is merely a void or vacancy in a surrounding substance, then properly speaking is the hole a thing in itself? Philosopher Roberto Casati writes, “Ask any person to tell you what holes are — ‘real,’ everyday holes, not the abstract holes of geometry – and he will likely elaborate upon absences, nonentities, nothingnesses, things that are not there. Are there such things?”

A hole story from John Timbs’ 1873 *A Century of Anecdote*:

A gentleman, one Sunday morning, was attracted to watch a young country girl on the high road from the village to the church, by observing that she looked hither and thither, this way and that upon the road, as if she had lost her thimble. The bells were

settlingfor prayers, and there was no one visible on the road except the girl and the gentleman, who recognised in her the errand-maid of a neighbouring farmer. ‘What are you looking for, my girl?’ asked the gentleman, as the damsel continued to pore along the dusty road. She answered, gravely: ‘Sir, I’m looking to see if my master be gone to church.’ Now, her master had awooden leg.

In number theory, a Holey prime is a prime number made up exclusively of the digits 4, 6, 8, 9, and 0 — digits whose Arabic numerals contain “holes.” Ironically, these get pretty substantial: The largest known specimen is a 4 followed by 16,131 9s.

## Bewitched

What’s remarkable about these numbers?

They form a perfect magic square. Each row, column, and diagonal adds to 81.

W.S. Andrews wrote, “Considering its constructive origin and interesting features, this square, notwithstanding its simplicity, may be fairly said to present one of the most remarkable illustrations of the intrinsic harmony of numbers.”

## The Deconstructed Self

Someone in whose power I am tells me that I am going to be tortured tomorrow. I am frightened, and look forward to tomorrow in great apprehension. He adds that when the time comes, I shall not remember being told that this was going to happen to me, since shortly before the torture something else will be done to me which will make me forget the announcement. This certainly will not cheer me up, since I know perfectly well that I can forget things, and that there is such a thing as indeed being tortured unexpectedly because I had forgotten or been made to forget a prediction of the torture: that will still be a torture which, so long as I do know about the prediction, I look forward to in fear. He then adds that my forgetting the announcement will be only part of a larger process: when the moment of torture comes, I shall not remember any of the things I am now in a position to remember. This does not cheer me up, either, since I can readily conceive of being involved in an accident, for instance, as a result of which I wake up in a completely amnesiac state and also in great pain; that could certainly happen to me, I should not like it happen to me, nor to know that it was going to happen to me. He know further adds that at the moment of torture I shall not only not remember the things I am now in a position to remember, but will have a different set of impressions of my past, quite different from the memories I now have. I do not think that this would cheer me up, either. For I can at least conceive the possibility, if not the concrete reality, of going completely mad, and thinking perhaps that I am George IV or somebody; and being told that something like that was going to happen to me would have no tendency to reduce the terror of being told authoritatively that I was going to be tortured, but would merely compound the horror. Nor do I see why I should be put into any better frame of mind by the person in charge adding lastly that the impressions of my past with which I shall be equipped on the eve of torture will exactly fit the past of another person now living, and that indeed I shall acquire these impressions by (for instance) information now in his brain being copied into mine. Fear, surely, would still be the proper reaction: and not because one did not know what was going to happen, but because in one vital respect at least one did know what was going to happen — torture, which one can indeed expect to happen to oneself, and to be preceded by certain mental derangements as well. If this is right, the whole question seems now to be totally mysterious.

– Bernard Williams, *Problems of the Self*, 1973

## Odd and Even

Well, it’s our old friend the mysterious pouch. Today the pouch contains a random quantity of marbles, and we’re going to withdraw a handful. But first, consider:

- If the bag contains an even number of marbles, then we are equally likely to withdraw an even or an odd number. For instance, if it contains 4 marbles, then we are equally likely to withdraw 2 or 4 as 1 or 3.
- But if the pouch contains an odd number of marbles, then we’re more likely to withdraw an odd number, as there’s one more way of choosing an odd number than an even number. For example, if the pouch contains 5 marbles then we’re more likely to draw 1, 3, or 5 than 2 or 4.

This is troubling. Without even opening the pouch we seem to have decided that, on balance, we’re more likely to withdraw an odd number of marbles than an even. Indeed, this seems to mean that handfuls *in general* are more commonly odd than even. How can this be?