Science & Math
Hands Up?
Beginning poker players are often shown a table like this:
It’s straightforward enough, assigning a hierarchy to the hands based on the likelihood of their appearance. But a strange thing happens when wild cards are introduced. Suppose we add one wild joker:
Now three of a kind is more likely (and thus less valuable) than two pair. Well, can we just reverse their places in the table? No, we can’t, because the wild card permits some players to reinterpret their hands. If you’re holding 6♠ 6♥ 7♣ 10♦ plus the joker, and we change the table, you’ll simply decide you’re holding two pair rather than three of a kind. So will everyone in your position. In fact, if we recalculate the odds with this expectation, we find that two pair has again become the more likely hand (13:1 vs. 34:1).
This can go on all day. Whenever a hand is declared “rare” it becomes popular — and thus not rare. The bottom line is that when wild cards are allowed, it becomes impossible to rank hands based on frequency.
From Julian Havil, Impossible?, 2008.
Numbers Game
On June 18, 1964, an elderly woman was walking through a Los Angeles alley when a blond woman with a ponytail pushed her to the ground and stole her purse. The blond woman escaped in a yellow car driven by a bearded black man.
Police arrested Janet Collins, a ponytailed blond woman whose bearded black husband drove a yellow Lincoln. At trial, a local mathematics instructor testified that there was 1 chance in 12 million that another couple would meet this description, and the jury convicted the Collinses of second-degree robbery. Sound reasonable?
Well, no. The California Supreme Court reversed the conviction, noting that the prosecution had offered no statistical evidence and that the mathematician had simply invented estimates for each of the six factors and multiplied them together, without adjusting for dependence or the possibility of mistake.
“The testimony as to mathematical probability infected the case with fatal error and distorted the jury’s traditional role of determining guilt or innocence according to long-settled rules,” wrote justice Raymond Sullivan. “Mathematics, a veritable sorcerer in our computerized society, while assisting the trier of fact in the search for truth, must not cast a spell over him.”
Sound Reasoning
Wintering in the Canadian Arctic in 1822, Capt. W.E. Parry made a series of experiments to see whether cold affects the velocity of sound. He marked a line of 5,645 feet on the sea ice, put a six-pounder gun at one end, and stood with a second observer at the other end. The gun fired 15 blank charges, and the observers timed the interval between each flash and its report. Generally they got good results, giving a mean velocity of 1,023 feet per second. But, writes Mr. Fisher:
The Experiments on the 9th February, 1822, were attended with a singular circumstance, which was–the officers’ word of command ‘fire,’ was several times distinctly heard both by Captain Parry and myself, about one beat of the chronometer [half a second] after the report of the gun; from which it would appear, that the velocity of sound depended in some measure upon its intensity.
“The word ‘fire’ was never heard during any of the other experiments; upon this occasion the night was calm and clear, the thermometer 25° below zero, the barometer 28.84 inches, which was lower than it had ever been observed before at Winter Island.” The phenomenon, whatever it was, has never been observed elsewhere, but Parry noted another acoustic oddity on his next voyage.
A Chemical Compound
What’s unusual about this list of elements?
- Protactinium
- Radium
- Praseodymium
- Oxygen
- Iron
- Sulfur
- Silicon
- Oxygen
- Nitrogen
- Aluminum
- Sulfur
Assemble their symbols and you get PaRaPrOFeSSiONAlS.
Other long “chemistry words”: HYPoThAlAmICoHYPoPHYSeAlS and PNEuMoCYSTiS CArInII PNEuMoNiAs.
Round Numbers
In 1986 The Mathematical Intelligencer published this story about devising a mnemonic for a famous constant:
For a time I stood pondering on circle sizes. The large computer mainframe quietly processed all of its assembly code. Inside my entire hope lay for figuring out an elusive expansion. Value: pi. Decimals expected soon. I nervously entered a format procedure. The mainframe processed the request. Error. I, again entering it, carefully retyped. This iteration gave zero error printouts in all–success. Intently I waited. Soon, roused by thoughts within me, appeared narrative mnemonics relating digits to verbiage! The idea appeared to exist but only in abbreviated fashion–little phrases typically. Pressing on I then resolved, deciding firmly about a sum of decimals to use–likely around four hundred, presuming the computer code soon halted! Pondering these ideas, words appealed to me. But a problem of zeros did exist. Pondering more, solution subsequently appeared. Zero suggests a punctuation element. Very novel! My thoughts were culminated. No periods, I concluded. All residual marks of punctuation = zeros. First digit expansion answer then came before me. On examining some problems unhappily arose. That imbecilic bug! The printout I possessed showed four nine as foremost decimals. Manifestly troubling. Totally every number looked wrong. Repairing the bug took much effort. A pi mnemonic with letters truly seemed good. Counting of all the letters probably should suffice. Reaching for a record would be helpful. Consequently, I continued, expecting a good final answer from computer. First number slowly displayed on the flat screen–3. Good. Trailing digits apparently were right also. Now my memory scheme must probably be implementable. The technique was chosen, elegant in scheme: by self reference a tale mnemonically helpful was ensured. An able title suddenly existed–“Circle Digits”. Taking pen I began. Words emanated uneasily. I desired more synonyms. Speedily I found my (alongside me) Thesaurus. Rogets is probably an essential in doing this, instantly I decided. I wrote and erased more. The Rogets clearly assisted immensely. My story proceeded (how lovely!) faultlessly. The end, above all, would soon joyfully overtake. So, this memory helper story is incontestably complete. Soon I will locate publisher. There a narrative will I trust immediately appear, producing fame. THE END.
The text explains itself: Count the letters in each word (a punctuation mark other than a period counts as a 0, and a digit stands for itself), and you’ll get the first 402 digits of π.
A Close Shave
Astronomers have the light-year, but nuclear physicists need an analogous unit for measuring tiny distances.
Happily, they have one: The Physics Handbook for Science and Engineering defines the “beard-second” as the length the average physicist’s beard grows in one second, or about 5 nanometers.
Google will even make the conversion for you — type 1 inch in beard-seconds into your search box and see what you get.
Tempting
The Earl of Yarborough offers you a wager. He’ll shuffle an ordinary deck and deal you 13 cards. If none of your cards ranks above 9, he’ll give you a thousand pounds. Otherwise you must give him one pound.
Should you accept?
The Horizontorium
This clever anamorphic illusion was invented by W. Shires in 1821. Cut out the center piece, make a hole at A, fold it at B, and position it at D. (Here’s a larger version.)
Peer through the hole with one eye, preferably with a light source on your right, and you’ll see the tombstone in three dimensions, surrounded by a low palisade.
Here’s another scene using the same principle; position the eyepiece where the turrets’ lines would converge and “the whole view will appear in its just proportions, representing a castle at a considerable distance, the loftiest part of which appearing scarcely an inch high.”
“Calculation and Memory”
William Lawson, teacher of mathematics in Edinburgh, who died in 1757, when employed about twenty years before his death as preceptor to the sons of a gentleman, was induced by his employer to undertake an extraordinary piece of mental calculation. Upon a wager laid by his patron, that the numbers from 1 to 40 inclusive could, by memory alone, be multiplied continually–that is, 1 multiplied 2; the product then arising, 2, by 3; the next product, 6, by 4; the next, 24, by 5; and so on, 40 being the last multiplier–Mr. Lawson was, with reluctance, prevailed upon to attempt the task. He began it next morning at seven o’clock, taught his pupils their Latin lessons in the forenoon as usual, had finished the operation by six in the evening, and then told the last product to the gentlemen who had laid the wager; which they took down in writing, making a line of forty-eight figures, and found to be just. … When the operation was over, he could perceive his veins to start, like a man in a nervous fever; the three following nights he dreamed constantly of numbers; and he was often heard to say that no inducement would ever again engage him in a like attempt. A fair copy of the whole operation, attested by the subscriptions of three gentlemen, parties in the wager, was put into a frame with glass, and hung up in the patron’s dining-room.
— Chambers’s Journal, Sept. 27, 1856