Capacity

In the 1967 Star Trek episode “The Trouble With Tribbles,” a small furry alien species is introduced on board the Enterprise and after three days grows to 1,771,561 individuals. In 2019 University of Leicester physics undergraduate Rosie Hodnett and her colleagues wondered how long it would take for the creatures to fill the whole starship. Using Mr. Spock’s estimate that each tribble produces 10 offspring every 12 hours and assuming that each tribble occupies 3.23 × 10^-3 m³ and that the volume of the Enterprise is 5.94 × 10⁶ m³, they found that the ship would reach its limit of 18.4 × 10⁹ tribbles in 4.5 days.

A separate inquiry found that after 5.16 days the accumulated tribbles would be generating enough thermal energy to power the warp drive for 1 second.

(Rosie Hodnett et al., “Tribbling Times,” Journal of Physics Special Topics, Nov. 18, 2019.)

June 11, 2024June 10, 2024 | Entertainment · Science & Math

Unquote

“When you praise someone you call yourself his equal.” — Goethe

June 10, 2024June 9, 2024 | Quotations

Helmet Crash

A problem proposed by Mel Stover for the April 1953 issue of Pi Mu Epsilon Journal:

After a meeting of six professors, each man left with another’s hat. The hat that Aitkins took belonged to the man who took Baily’s hat. The man whose hat was taken by Caldwell took the hat of the man who took Dunlop’s hat. And the man who took Easton’s hat wasn’t the one whose hat was taken by Fort. Who took Aitkins’ hat?

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This solution is by Leon Bankoff. Denote each man by his initial and write x → y to indicate that x’s hat was taken by y. We know that B → x → A, where x is the man who took Baily’s hat, and that D → z → y → C, where z is the man who took Dunlop’s hat and y’s hat was taken by Caldwell. These two sequences comprise seven symbols altogether, so x, y, or z must be duplicating A, B, C, or D in one of them. There are two ways to combine the sequences: B → D → A → y → C (where x = D and z = A) D → z → B → C → A (where x = C and y = B) Each of these new sequences has five symbols, so each lacks one. Add s to the first sequence and t to the second to provide this missing link and to make each sequence into an unbroken cycle: s → B → D → A → y → C → s t → D → z → B → C → A → t The first cycle fits the problem statement if we let y = F and s = E, and the second matches if we let t = F and z = E. In either case, Fort took Aitkins’ hat. (“Problem Department,” Pi Mu Epsilon Journal 1:8 [April 1953], 329-330.)

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June 10, 2024June 9, 2024 | Puzzles

Feats

https://commons.wikimedia.org/wiki/File:Conquistador_Alonso_de_Ojeda.png — Images: Wikimedia
Commons

This just caught my eye — a feat attributed to the Spanish explorer Alonso de Ojeda, who would later accompany Columbus and name Venezuela:

Queen Isabella being in the tower of the cathedral at Seville, better known as the Giralda, Ojeda, to entertain her majesty, and to give proofs of his courage and agility, mounted on a great beam which projected in the air, twenty feet from the tower, at such an immense height from the ground, that the people below looked like dwarfs, and it was enough to make Ojeda himself shudder to look down. Along this beam he walked briskly, and with as much confidence as though he had been pacing his chamber. When he arrived at the end, he stood on one leg, lifting the other in the air; then turning nimbly round, he returned in the same way to the tower, unaffected by the giddy height, whence the least false step would have precipitated him and dashed him to pieces.

Allegedly he afterward threw an orange to the top of the tower, which would have been at least 88 meters tall at the time. This account is from Washington Irving’s biography of Columbus, and even Irving calls it “unworthy of record, but that it exhibits the singular character of the man.”

Emerson mentions Ojeda’s balancing act in his 1870 essay on success, claiming also that “Olaf, king of Norway, could run round his galley on the blades of the oars of the rowers when the ship was in motion.” I’m not even pursuing that one.

06/12/2024 UPDATE: Apparently oar-walking is not as far-fetched as it sounds. While the feat is rare, it is said that King Olafr Tryggvason did indeed run on the oar blades of his ship the Long Serpent as it was being rowed. Kirk Douglas manages the feat pretty well in the 1958 film The Vikings (thanks, Orion):

June 9, 2024June 12, 2024 | Oddities

Double Duty

https://commons.wikimedia.org/wiki/File:Clock_10-10.svg — Image: Wikimedia Commons

I just came across this arresting sentence in The Satanic Verses, of all places:

“Turn your watch upside down in Bombay and you see the time in London.”

It appears this is roughly true: Because Indian Standard Time has an offset of UTC+05:30, an analog watch set to Indian time and read upside down will give the time in London — 10:10 becomes 4:40, noon becomes 6:30, and so on. The reverse is also true — a London watch read upside down will give the time in India.

Unfortunately the hand positions are only approximate, and the U.K. observes daylight saving time and India doesn’t, so just now it doesn’t work. Interesting idea, though.

06/16/2024 UPDATE: Reader Kieran Child points out also that the trick cannot work perfectly as described as we need to add 5 hours 30 minutes in one direction and 6 hours 30 minutes in the other. “By studying it for a while, you will see that going from UK time to Indian time only works when the minutes are between 31 and 59, and going the other way only works when the minutes are between 00 and 29. For times outside of these ranges, you will be off by one hour.” Examples are sometimes chosen to conceal this confusion. (Thanks, Kieran.)

June 8, 2024June 16, 2024 | Oddities

Assimilation

He that is nourished by the acorns he picked up under an oak, or the apples he gathered from the trees in the wood, has certainly appropriated them to himself. Nobody can deny but the nourishment is his. I ask, then, when did they begin to be his? when he digested? or when he eat? or when he boiled? or when he brought them home? or when he picked them up?

— John Locke, An Essay Concerning the True Original, Extent and End of Civil Government, 1689

June 7, 2024 | Oddities

Shining Sea

Early European colonists were staggered at the abundance of fish in the Chesapeake Bay. William Byrd II wrote in his natural history of Virginia:

Herring are not as large as the European ones, but better and more delicious. When they spawn, all streams and waters are completely filled with them, and one might believe, when he sees such terrible amounts of them, that there was as great a supply of herring as there is water. In a word, it is unbelievable, indeed, indescribable, as also incomprehensible, what quantity is found there. One must behold oneself.

More accounts here. “The abundance of oysters is incredible,” marveled Swiss explorer Francis Louis Michel in 1701. “There are whole banks of them so that the ships must avoid them. A sloop, which was to land us at Kingscreek, struck an oyster bed, where we had to wait about two hours for the tide.”

June 7, 2024June 6, 2024 | History

Self-Seeking

There is nothing contradictory in imagining causal chains that are closed, though the existence of such chains would lead to rather unfamiliar experiences. For instance, it might then happen that a person would meet his own former self and have a conversation with him, thus closing a causal line by the use of sound waves. When this occurs the first time he would be the younger ego, and when the same occurrence takes place a second time he would be the older ego. Perhaps the older ego would find it difficult to convince the younger one of their identity; but the older ego would recall an identical experience long ago. And when the younger ego has become old and experiences such an encounter a second time, he is on the other side and tries to convince some ‘third’ ego of their physical identity. Such a situation appears paradoxical to us; but there is nothing illogical in it.

— Hans Reichenbach, The Direction of Time, 1956

June 6, 2024June 5, 2024 | Oddities

The Biter Bit

Amsterdam artist Hendrik de Keyser’s 1615 sculpture Screaming Child Stung by a Bee was probably inspired by an idyllium of the Greek poet Theocritus in which Cupid is stung while stealing honey from a hive. When he complains to his mother that so small a creature should cause such great pain, she responds that he himself is small but deals wounds that are grievous.

Painter Joseph Ducreux and sculptor Franz Xaver Messerschmidt also experimented with extreme facial expressions.

June 6, 2024June 5, 2024 | Art

Catch 22

From reader Chris Smith:

Pick a three-digit number in which all the digits are different. Example: 314.

Now list every possible combination of two digits from the chosen number. In our example, these are 13, 14, 31, 34, 41, and 43.

Divide the sum of these two-digit numbers by the sum of the three digits in the original number, and you’ll always get 22. In our example, (13 + 14 + 31 + 34 + 41 + 43) / (3 + 1 + 4) = 176/8 = 22.

This works because 10a + b, 10a + c, 10b + a, 10b + c, 10c + a, and 10c + b sum to 22a + 22b + 22c = 22(a + b + c), so dividing by a + b + c will always give 22.

(Thanks, Chris.)

06/08/2024 Reader Tom Race points out that essentially the same trick can be performed using the entire number: If you add all six permutations of the original 3 digits, then divide that total by the sum of the 3 digits, the answer is always 222.

For example, using 561:

561 + 516 + 156 + 165 + 651 + 615 = 2664

5 + 6 + 1 = 12

2664 / 12 = 222

“This works because in the first sum each of the three digits (a, b and c) occurs twice in each of the three columns, so the sum is 222a + 222b + 222c = 222(a + b + c).” (Thanks, Tom.)

June 5, 2024June 8, 2024 | Science & Math