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Trivia

Homo sum, humani nihil a me alienum puto.
I am a human being,  I think nothing human is alien to me.
Terence [PubliusTerentius Afer (c.190-159 BC) former Phoenician slave]
 Michon
 

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"Puzzles"[trivia, really] byJohn Baez.
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What's All ThisMeasurement Stuff, Anyhow?   by Robert A. Pease.
 
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Trivia


(2003-12-03)  

There are two  unsolved mathematical problemswhich date backat leastto the times ofEuclid (c.325-275 BC).  Namely:

  • Find an odd perfect number,or prove that none exist. 
  • Are there infinitely many twin primes?
    (Twin primes are pairs ofprime numbers whose difference is 2.)

A positive answer to the latter question is the Twin Prime Conjecture.


(2003-06-13)  
How is the north-south polarity of magnets defined?  What about Earth?

 William Gilbert  (1544-1603) People have always called north  the part ofa small magnet (or compass needle) that points northward (westill do).

The north pole of a magnet seeks the south pole of another... The idea was first proposed by SirWilliam Gilbert(1544-1603) [William Gylberde of Colchester] inDe Magnete(1600) that the Earth is like a giant magnetwhose magneticsouth is somewhere upnorth... Nevertheless, the common (misleading) usage is to keep calling"Magnetic North Pole" or, better, "North Magnetic Pole"(best capitalized) whichever magnetic pole of the Earth is nearitsgeographic north pole, although this is, technically, thesouth side of theEarth's magnetic field.

 Direction of Magnetic Field (B)  

The magnetic poles of the Earth are notoriouswanderers. In May 2001, theNorthMagnetic Pole was located at  81.3°N  110.8°W,moving northwest in the Canadian Arcticat about 40 km per year (more than 1 mm/s). If theRoss coat of arms is to be believed,the Magnetic Pole was first found on June 1, 1831, at 70.09°N  96.78°W. Thus, it moved a distance of about 11.65° of agreat arc in 170 years  (1295 km). For a straight motion, that would be an average of  7.6 km per year (21 meters a day,  87 cm per hour,  0.24 mm per second).

Over geological periods of time, the magnetic field of the Earth hasreversedits north-south polaritymany timeswith respect to the Earth's axis of rotation: It flips once every 400 000 years or so... The same thing happens with the magnetic field of the Sun, onlymuchmore frequently.


(2003-06-13)  
How theright-hand rule determines the sign ofaxial quantities.

North side of anelectrical circuit.South side of anelectrical circuit. 
North SideSouth Side
The mnemonic device pictured at left is usefulto identify thepolarity of the magnetic fieldcreated by a simple electrical circuit: Arrows drawn at the extremities of the capital letters "N" or "S" indicate thedirection of the current, when you're watching eithertheNorth orSouth side of the circuit. This engineering convention is (fortunately) compatible with thetraditional definition of the north pole of a magnet,as describedabove.

Rotating Earth

Happily, the above mnemonic trick also describes the waythe Earth rotates(counterclockwise, if you are above the geographicnorth pole). By analogy,astronomers andothers callnorth the side of a rotating celestial bodyfrom which it is seen to rotate counterclockwise(regardless of whatever magnetic field may prevail around such a body).

The direction of the right thumb indicatesa rotation pointed to by the other fingers.Seen from 'above', a positiverotation is counterclockwise.Mathematically, a rotation is apseudovector (whosemagnitude is the angular speed   ) directednorthward along the axis of rotation. This states the same convention as the so-called "right hand rule": Fingers of aright hand show the direction of motionin a rotation whose vector is in the direction of the thumb.

The prefixpseudo- is often used to denote a quantity which isso defined that itssign would change if our notions of"left" and "right" where reversed (the rotation pseudovector is one example). Such quantities may also be calledaxial,in contradistinction to the more familiarpolar quantities,whose definitions donot depend on any particular conventionconcerning "space orientation".

The cross-product of two [polar] vectors is axial, so is the cross-productof two axial vectors.  However, the cross-product of a polar vector anda pseudovector ispolar. For example, the relation V = R holds, which gives thevelocityV [a polar vector]for a point of a rotating solid,if is the [axial]vector of rotation,andR is that point's position [another polar vector]with respect to some motionless origin located anywhere on the axis of rotation...


(2003-12-02; anonymous query)   What initiates the wind?

A great question. Several unrelated primitive cultures had a similar answer:

It's the "tail" of the Sun.

Well, they were right... Meterological details may be extremely complex, butthe wind is caused by differences in pressure whichultimately come fromheat produced by the Sun's rays.

On coastlines for example, breezes are felt up to 50 km inland (landward during the daytime and seaward at night) which are due to the fact that the Sun's radiation changesthe temperature of land masses faster than it changes thetemperature of the Ocean. As a result, the land is warmer than the sea by day, and colder by night.

Global and large scale patterns are also due to the Sun but are less obvious:

Patternof Prevailing Winds at the Surface of the Globe
NameLatitudePrevailing Direction
 North PoleVariable (high pressure)
Polar EasterliesNorthern CircumpolarFrom NE to SW
 NorthernVariable (low pressure)
WesterliesMidnorthernFrom SW to NE
Horse LatitudesAround 30°NVariable or calm (high pressure)
Northeast Trade WindNorthern TropicalFrom NE to SW
DoldrumsEquatorialVariable or calm (low pressure)
Southeast Trade WindSouthern TropicalFrom SE to NW
Horse LatitudesAround 30°SVariable or calm (high pressure)
Roaring FortiesMidsouthernFrom NW to SE
 SouthernVariable (low pressure)
Polar EasterliesSouthern CircumpolarFrom SE to NW
 South PoleVariable (high pressure)

A dubious legend surrounds the naming of "horse latitudes": A sailboat could be becalmed there for so long that the horses it carried wouldhave to be thrown overboard, in order to save fresh water for people...

The term "wind" is best reserved for the flow of air near the surface of the Earth. At higher altitudes, the term "current" is preferred. The prevailing high-altitude currents always blow from the west (the strongest such steady current is known as the Jet Stream). The prevailing easterly winds  (the tropical trade winds and the polar easterlies) are thus relatively shallow...

The direction and intensity of the wind is often dominated by travelling cyclonic oranticyclonic meteorological disturbances (especially in the zones indicated as "variable" in the above table).

cyclone  is alow pressure zone, around which windsrotate counterclockwise in the northern hemisphere (clockwise in the southern hemisphere).

An anticyclone  is ahigh pressure zone, around which windsrotate clockwise in the northern hemisphere (counterclockwise in the southern hemisphere).


kokapelli (Indianapolis, IN.2000-09-07)
In algebra, why is the letter "m" a symbol for slopes in linear equations?
prisonin (2000-11-10)
Why is slope "m", in the "slope-intercept" form of a linear function?
1621 (2002-04-29)
Why is "m" used for slopes?     [As in:   y =m x  + b ]
BooBooLuvsU (2002-04-02)
Why is "m" used to represent the slope in a linear equation?

Well, the explanation is certainlynot the one most often given,namely that "m" is the first letter of the French verb "monter",meaning "to climb";I happen to know first-hand that virtuallyall French textbooksquote the generic linear function as  y = ax+b. If the tradition was of French origin, wouldn't the French use it?

In anearlier forumon this [apparently popular] subject, John H. Conway (1937-2020) rightly called the above explanation an "urban legend". He half-heartedly put forward [and later half-heartedly recanted]the theory that what we now call "slope" was once better known as "modulus of slope"("modulus of..." has often been used to mean "the parameter which determines..."). In 1990, Fred Rickey (of Bowling Green University, OH) could not even findany usebefore 1850 of the word "slope" itself to denote the tangent of a line's inclination...

Conway "seemed to recall" that Euler (1707-1783) did usem for slope,which remains unconfirmed. However, Dr. Sandro Caparrini (University of Torino) found outthat at least onecontemporary of Euler did so, sinceVincenzoRiccati (1707-1775) used the notationy =mx+nas early as 1757, in a reference toJakobHermann (1678-1733). (This and other related facts have been reported onlinein the excellent historical glossary ofJeff Miller;look underSlope.)

Eric Weisstein reportsthat the use of the symbolm for a slope was popularized around 1844[A Treatise on Plane Co-Ordinate Geometry, by M. O'Brien. Deightons (Cambridge, UK) 1844]and subsequently through several editions of a popular treatise by Todhunter,whose notation wasy =mx+c.[Treatise on Plane Co-Ordinate Geometry as Applied tothe Straight Line and the Conic Sectionsby I. Todhunter, Macmillan (London, UK) 1888].

The preferred notation for the slope-intercept cartesian equation ofa straight line in the plane isnot at all universal. Here's what I have gleaned so far:

 y =m x  + nVincenzo Riccati (1757). Netherlands, Uruguay.
 y =m x  + cUK.
 y =m x  + bUS, Canada.
 y =a x  + bFrance,Netherlands, Uruguay.
 y =k x  + bRussia.
 y =k x  + mSweden.
 y =k x  + dAustria.
 y =p x  + qNetherlands.

Please,let me knowif you are in a position to add to the above table(also if you can confirm or invalidate any part of it).  Thanks.

Acknowledgements :

Information for Uruguay and Netherlands due to Julie Budnik (2005-12-19).


(Bob of Sacramento, CA.2000-12-04)  
What is the mark at  19"3/16  on a tape measure? It repeats itself.


The so-called diamond mark  is actually positioned at exactly 8/5 of a foot (that's exactly 1.6' or 19.2 inches, which is indeed pretty close to19 3/16 ).

The diamond marks are also called "black truss" markings, because they correspondto the truss layout which is used with 8-foot sheets of plywood(or other material), namely 5 trusses per sheet.

This is to be contrasted with "red stud" markings which appearevery 16 inches by showing the corresponding inch number in red instead of black.The black markings and the red markings coincide at 8-foot intervals (96 inches). That's to say:  5 black intervals or 6 red ones in an 8-foot width.

5/8 = 0.625 is a standard slope for a roof, which may thus be builtby measuring horizontally as manydiamonds as there are vertical feet.

 = (1+5)/2, which isabout 1.618034.

den0eng3 (2002-06-28)
What is the largestExcel expression of at most 35 keystrokes?

Excelinterprets something like 3^3^3 as (3^3)^3 = 27^3 = 19683. (For some calculators, this expressionmeans3^(3^3) = 3^27 = 7625597484987.)

This idiosyncrasy of Excel  makes the question interesting, because of theparentheses needed to make a tower  of exponents. (Otherwise, the answer would simply be 9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9E9.)

Largest number  Nn expressible in n keystrokes or less.
N19
N299
N39E9 has 10 digits.
N49E99 has 100 digits.
N59^9E9 has 8588182585 digits.
N69^9E99 has over 8.588 digits.
N79^9E999 has over 8.588 digits.
N89^9E9999 has over 8.588 digits.
N99^(9^9E9) has over 0.95424 N5 digits.
N109^(9^9E99) has over 0.95424 N6 digits.
N119^(9^9E999) has over 0.95424 N7 digits.
N129^EXP(9^9E9) has over 0.95424 exp(N5) digits.
N139^EXP(9^9E99) has over 0.95424 exp(N6) digits.
Nn+79^EXP( )
N349^EXP(9^EXP(9^EXP(9^EXP(9^9E99))))
N9^EXP(9^EXP(9^EXP(9^EXP(9^9E999))))
N369^EXP(9^EXP(9^EXP(9^EXP(9^9E9999))))
N379^EXP(9^EXP(9^EXP(9^EXP(9^(9^9E9)))))
N389^EXP(9^EXP(9^EXP(9^EXP(9^(9^9E99)))))

A nice function to generate large numbers is thefactorialfunction.  However, itsExcel name (FACT) is longer than thatof the exponential function (EXP) which is thus allowed to win the day for N12,in spite of its less extreme growth.

What happens next is illustrated by the N13 case,where we had to choose the largest candidate among 9^EXP(9^9E99)9^FACT(9^9E9),  and possibly9^(9^(9^9E9))... Because the argument of EXP has one more keystroke, the correspondingexpression is the largest by a wide margin.  The somewhat largergrowth of FACT does not help much, in this case or in any subsequent one...

For 12 keystrokes or more, the largest expression is"9^EXP(...)" with aninner expressionfound 7 steps before in the table (7 fewer keystrokes). This makes the table extremely easy to extend beyond the 12th entry,and we may quickly obtain the final answer to the original question(35 keystrokes):

9^EXP(9^EXP(9^EXP(9^EXP(9^9E999))))

Only functions with exponential growth and shorter names,like CH or SH, could possibly allow this record to be broken,but inExcel such potential candidates have longer names which rules them out(namely, COSH and SINH).


For a larger number of keystrokes, the following technique may or may not be acceptable (as it's a pureExcel idiosyncrasy, not shared by similar languages). The basic idea is to use compact descriptions of extremely long textstrings (representing syntactically correct numbers) using the functionREPT(x,n),which returns n concatenated copies of whatever string is specified by x. Such strings can be explicitly converted into syntactically acceptable numbersusing the EVALUATE function (which works if macros are allowed). Either one of the following related patterns will thus convert a numericalexpression of length n intoan expression of length 2n+35 or 2n+36 representing atower of exponents. Which pattern you use depends on the parity of the allowed number of keystrokes,whenever that number is 39 or more. The method doesn't apply to less than 37 keystrokes and is inferior to the abovefor 37 or 38 keystrokes.

EVALUATE(REPT("9^(",)&9&REPT(")",))EVALUATE(REPT("9^(",)&99&REPT(")",))

The operator "&" is used to concatenate strings. Quotes around the inner "9" or "99"are not needed since integers are converted to strings whenever appropriate.Thanks toden0eng3for suggesting the use of EVALUATE...

EVALUATE(REPT("9^(",99)&9&REPT(")",99))
Largest number  Nn expressible in n keystrokes (continued).
N39
N40EVALUATE(REPT("9^(",99)&99&REPT(")",99))
N41EVALUATE(REPT("9^(",9E9)&9&REPT(")",9E9))
N42EVALUATE(REPT("9^(",9E9)&99&REPT(")",9E9))
N43EVALUATE(REPT("9^(",9E99)&9&REPT(")",9E99))
N44EVALUATE(REPT("9^(",9E99)&99&REPT(")",9E99))
N43EVALUATE(REPT("9^(",9^9E9)&9&REPT(")",9^9E9))
N44EVALUATE(REPT("9^(",9^9E9)&99&REPT(")",9^9E9))
N2n+35EVALUATE(REPT("9^(",)&9&REPT(")",))
N2n+36EVALUATE(REPT("9^(",)&99&REPT(")",))

Finding what string of given length describes the largest number in a given formallanguage is a variant of what's known as theBusy Beaver Problem (the name comes from the impression you get from watching small Turing machinesgenerate large outputs). For a general enough computer language(i.e., anything as powerful as a Turing machineno  algorithm could possibly solve this problem! A single Excel  expression falls short of that intractable category,but a whole Excel  spreadsheet would be in it...


WiteoutKing (Lowell, MA.2002-07-17)   [See alsounabridged answer.]
What are theodds in favor of being dealt a givenpoker hand?

There are C(52,5) = 2598960 different poker hands and each of them is dealt with thesame probability. [See detailselsewhere on this site.]

Theprobability of a given type of hands is thus the number of such handsdivided by 2598960. When the probability of something is the fractionP = x / (x + y),its so-calledodds are said to be eitherx to y in favorory to x against, as shown in the table below,which assumes somefamiliarity with poker (10 kinds of "straights" are normally allowed,see below orhere).

TypeNumber of HandsProbabilityOdds in Favor
Royal FlushC(4,1) C(1,1)1 / 6497401 to 649739
Straight FlushC(4,1) C(10-1,1)36 3 / 2165803 to 216577
4 of a KindC(13,1) C(48,1)624 1 / 41651 to 4164
Full House13 C(4,3) 12 C(4,2)3744 6 / 41656 to 4159
FlushC(4,1) [C(13,5) -10]5108 1277/6497401277 to 648463
StraightC(10,1) (45-4)10200 5 / 12745 to 1269
3 of a Kind13 C(4,3) C(12,2) 4254912 88 / 416588 to 4077
Two PairsC(13,2) C(4,2)244123552 198 / 4165198 to 3967
Pair13 C(4,2) C(12,3) 431098240 352 / 833352 to 481
High Card(C(13,5)-10) (45-4)1302540 1277 / 25481277 to 1271
TOTALC(52,5)2598960 11 to 0

In the last entry,"High Card" means a hand that'snone of the above : Two such hands would be comparedhighest card first to decide who wins.

Note that there are normally 10 different "heights" for astraight andthat theace (A) belongs to the lowest (A,2,3,4,5) and the highest (10,J,Q,K,A),which is traditionally called aRoyal Flush if all cards belong to the same suit. Should your own local rules disallow the tenthstraight sequence(A,2,3,4,5), the tabulated counts for straights and/or flushes should be changed(and the "High Card" count should be modified as well),replacing 4, 36, 5108, 10200 and 1302540 respectivelyby 4, 32, 5112, 9180 and1303560 = (C(13,5)-9)(45-4).


(A.C. of Houston, TX.2001-02-12)
What are the numbers in reverse sequence on the verso of a book's title page,below the publication date?

The last of them indicates the number of the printing run for the copy you're holding:"10 9 8 7 6 5" means "fifth printing".There's also a similar sequence of double digits which indicates thedate of printing:The sequence "02 01 00 99 98" means "98", which would most probably be 1998,since I do not think [I may be wrong on this]that the system was in force in 1898 or earlier.In practice, the system would remain unambiguous even in the distant future, sincethe latest date appearing elsewhere on the page can't possibly predate the actual year ofprinting by more than a century...Please,let me know if you haveany information about approximatelywhen this practice started. Thanks.

The reason for this strange convention is quite practical:It allows the same plates to be used for each printing;the last number is simply carved out as needed for a new printing,so that it no longer appears on the paper.This saves time and money with traditional printing.


(2002-07-13)
How many living species are there on Earth?

Approximately 1400 000 species have been recognized,but the total number of species is estimated to beat least10 000 000. 

However, forall we know,the actual number could be as high as 100 million. There is currently no central database established by a recognized authority,although this may change with ongoing or future efforts,like theSpecies 2000 project,theALL Species Foundation, or theCensus of Marine Life(CoML).

In his 1992 bookThe Diversity of Life (Harvard University Press),Edward O. Wilson quotes a total number of 1402 900identified species. This inventory includes 751 000 insects, 123 400 noninsect arthropods, 106 300other invertebrates, and only 42 300 vertebrates(less than 10% of which are mammalian). The remainder consists of248 400 plants, 69 000 fungi, 57 700 protists(inluding 26 900 phototropic algae),and 4800 bacteria (the bacterial world is almost uncharted,seebelow). Wilson himself believes the actual total number of species alive on Earth to be"somewhere between 10 and 100 million". This seems to be the most often quoted range, although the Oxford specialist Robert M. Mayoffers a much lower guess of 5 to 8 million.

Wilson estimates that about 27 000 species disappear each year (about 3 per hour),mostly because of the eradication of the rain forest. This amounts to 1000 or 10 000 times the "natural" extinction rate prevalent inprehistoric times. In his 1994 bookVital Dust (BasicBooks),1974 Nobel laureate Christian de Duve quotes all ofthe above and calls thisthe biological equivalent of the burning of thelibrary of Alexandria in 641.

Even if cataloguing them would essentially be an endless task,the number ofbacterial species in a given sample can beestimatedstatistically by measuring only the total population and the number ofindividuals in the most prominent species. Dr. Tom Curtis (and his coworkers at theUniversity of Newcastle upon Tyne)did just that in a recent articleof theProceedingsof the National Academy of Sciences: A cubic centimeter of seawater typically holds about 160 species,and the entire ocean is expected to contain about 2000 000 distinct speciesof bacteria. On the other hand, a gram of garden soil harbors around 6300 species,and a ton may contain about 4000 000 of them. 


(2002-11-14)
What are the most primitive species still alive?

By studying genetic material at the molecular level (DNA),cladists are now able to obtain a fairly accurate picture of whatthe DNA of a group's common ancestor was like. They can also determine what species is closest to that ancestorand is thus the mostprimitive of them, probably because ithas been around the longest...  Examples includes:

 

(Cortney C. of Anacoco, LA.2000-10-10)
How many dimes are in an ounce? How many pennies are in an ounce?
(C. S. of Rayne, LA.2000-08-22)
When did the penny become a gram lighter?
(Anna of Rock Rapids, IA.2000-10-11)
What is the volume of a penny?

I assume you're talking aboutUS coins.

Pennies manufactured from 1793 to 1837 were pure copper.Before 1982, the penny was still almost a solid copper coin (95% copper, 5% zinc)and its nominal mass was set at 48 grains (about 3.11g).It was legally allowed to be as much as 2 grains above or below the nominalvalue, but practical tolerances were much tighter.

In the context of coins, thetroy ounce of 480 grains is more appropriate thanthe commonavoirdupois ounce of 437.5 grains. So your first answer is thatit takes 10 (pre-1982) US pennies to make atroy ounce(and about 9.1146 of these to make anavoirdupois ounce).

Since November 1982, the penny has been a copper-plated zinc coin(97.6% zinc, 2.4% copper) with a nominal mass of 2.5g[0.6 g lighter than before].

When the change from copper to zinc took place, the new pennies were engineered tohave the same look and size as the old ones. With such a nominal mass, the volume ofthe new penny was only reduced by 0.84% (see computation below). Since the nominaldiameter of the coin was held constant, this means itsthickness went downby 0.84%.

/°C at 25°C)so its density at 20°C is about(1 + )larger, or about 7.136 g/cc. We may assume --although that'snot quite true in practice-- that a gram of an alloy made from X grams of copper and(1-X) grams of zinc has the same volume as the total volume of the two componentmetals taken separately: X grams of pure copper (density d)on one side and (1-X) grams of pure zinc (densityd) on the other.With this assumption, the density of such an alloy is:1/(X/d+(1-X)/d).This makes the density of old pennies about 8.847 g/cc and that of new ones about7.171 g/cc, for a ratio of about 81.06%. As the old penny weighs a nominal48 grains (about 3.1103g), a new penny of the exact same size would weighabout 2.521 g. At a nominal mass of only 2.5 g, a new penny has therefore avolume which its about 0.84% smaller than the volume of an old penny...

Thevolume of a penny is very close to0.35 cc (0.0214 cubic inches);slightly more for a copper penny, slightly less for a zinc penny:The volume of a copper penny is about 0.3516 cc.This is obtained as the ratio of the nominal mass of a "copper" penny (3.1103 g)to the approximate density (8.847 g/cc) of the 95% copper alloy it is made of. Similarly, the volume of the "zinc" penny is about 0.3486 cc, the ratio of the newnominal mass (2.5g) to the new density of 7.171 g/cc(that's 0.84% less than the volume of a copper penny).

Dimes are about 2.268 g each. The nominal mass is 35 grains (2.26796185 g).With dimes, quarters, or half-dollars (see below), $20 worth of coins make anavoirdupois pound (7000 grains). There are 200nominal dimes in 7000 grains.

Isotopic Pennies.

The 20% difference in mass between pre-1982 and post-1982 US penniesis used as the basis of a classroom activity (known as "Isotopic Pennies") whichis meant to help chemistry students grasphow the average molar mass is related to the isotopic composition.For example, a single weighing of a stack of 10 pennies determines how manypre-1982 pennies are in it...



 


(Robert of Clifton, TX.2000-11-11)   
How many pennies are in a pound?
[How many pennies per avoirdupois pound?  US pennies in 1 lb.]

Assuming you're talking about US coins, there's a big problem: In November 1982,the US penny became about 0.6 gram lighter. The older coin was 95% copper and 5% zinc,while the new one is essentially copper-plated zinc (97.6% zinc and only 2.4% copper).The nominal mass of a penny before 1982 was 48 grains (about 3.11 g).The size of a penny changed very little (-0.84%) in 1982 but,because zinc is lighter than copper,the new coin's nominal mass is 2.5 g.

Before 1982, there was about 146 pennies in a pound... If all the pre-1982 pennies were out of circulation,there would be about 181 pennies to the pound.

Right now, a pound of pennies from the street will contain anywhere between146 and 181 pennies, depending on the percentage of pre-1982 pennies in it. Accordingto the US Mint, the approximate life span of a coin is about 25 years. If we take this number at face value, there remains in circulation today(November 2000; 18 years later) approximatelyexp(-18/25),or about 48.7% of the pennies that were in circulation in November 1982.

Assuming that the total number of pennies in circulation is the same todayas it was in 1982 (which is probably not quite true),this would mean that a penny's mass in grams averages about 3.11(0.487)+2.5(1-0.487)which is very close to 2.8 g, so that there would be just about162 pennies in a poundas of November 2000. If there's already more than 160 pennies in a pound,theaverage penny is already slightly less than 1/10 of anavoirdupois ounce!


 Packing Nickels samgiordano (2003-05-04; e-mail)
How much money is in five gallons of nickels?
[How many pennies, dimes or quarters per gallon?]

We're talking about US coins (5 cents) and USgallons(namelyWinchester gallons of exactly 231 cubic inches,which arevery different from Imperial gallons)...

We'll consider nickels to be perfect cylinders. Packing identical solids as densely as possibleis anotoriously difficult problem (only recently solved for spheres). For circular cylinders, we may guess that the solution involves optimal 2D layers (not necessarily aligned with each other) as illustrated above. Alternately, unlayered stacks of cylinders arranged next to each otherin this 2D hexagonal pattern fill 3D space with the same density. Denser packings do not seem possible, although we lack a rigorous proof of this"obvious" fact. This is how we may estimate the highestnumber ofnickels per gallon in large containers...

Thenominal diameter of a nickel is 0.835", or 21.209 mm(see31 USC 5112). TheUSMint online specificationsgive the thickness of a nickel as 1.95 mm,but I did a quickreality checkby measuring a stack of 20 nickelsand found it to be almost exactly 37 mm, instead of the expected 39 mm! Assuming a one-digittypo in the official site,we'll thus take the thickness of a nickel to be1.85 mm. The discrepancy is otherwise much too large to be attributed to normal wear... [Similar thickness measurements for other types ofcirculated coins matchthenominal data published online by the US Mint almost perfectly.]

In the aforementioned packing(s), each coin occupies (without voids or overlaps)the volume of a regular hexagon of thesame thickness circumscribed to it. The top surface area is ½3times the square of the coin'sdiameter. As there areexactly 25.4 mm to the inch, Hexagonal Prism the above numbers make this volume incubic inches equal to:

V   =  ½3 (0.835)2 (1.85/25.4)  =   0.0439786196844...

Since there areexactly 231 cubic inches in a US gallon, this translates into231/V, or about 5252.5523nickels per gallon. In 5 gallons, you'd haveat most 26262 nickels,worth $1313.10 (possibly a little bitmore if the walls of the containerare shaped to fit the coins, butmuch less in a random packing).

For pennies (diameter: 0.75", thickness: 1.55 mm) the above computation wouldgive a "V" of about 0.029727 cu in, or about 7771pennies per gallon.

Similarly, a gallon would contain [at best] roughly 10500 dimes, 4200 quartersor2100 half-dollars.  Any of these translates into $1050 per gallon. The result is the same for these 3 types of coins for a reasonyou have to figure out,but here's a clue:  Anymixture of these three types of coinsrepresents thesame amount of money, by weight. Seasoned geometricians would remark that a givenvolume would be lessvaluable with several coin types instead of a single one. All this, of course, is for theideal densest possible packing... In practice,YMWV.

On 2007-09-11,Bob Bernstein  wrote:
I just counted my very, very full to the rim gallon of pennies.
Ionly found  5612  pennies, instead of your estimate of  7771.

As advertised, the physical packing obtained by stuffing coins in a jarwill be substantially less dense than the densestpacking of cylinders described above. It's not at all surprising to find  27%  fewer coins in such a jarthan in an equivalent volume of neatly stacked coins.

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