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

(2003-12-03) What is the oldest open mathematical problem?

There are two unsolved mathematical problems
which date back at least
to the times of Euclid
(c.325-275 BC). Namely:

Find an oddperfect number,
or prove that none exist._{ }

Are there infinitely many twin primes?
(Twin primes are pairs of
prime numbers whose difference is 2.)

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

(2003-06-13) The "North Magnetic Pole" of the Earth...
How is the north-south polarity of magnets defined? What about Earth?

People have always called north the part of
a small magnet (or compass needle) that points northward
(we
still do).

The north pole of a magnet seeks the south pole of another...
The idea was first proposed by Sir
William Gilbert
(1544-1603) [William Gylberde of Colchester] in
De Magnete
(1600) that the Earth is like a giant magnet
whose magnetic south is somewhere up north...
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 near
its geographic north pole, although this is, technically, the south side of the
Earth's magnetic field.

In modern terms,
magnetic induction (B) is
defined so
that (loosely speaking) the
pseudovector B points away from the north
pole and toward the south pole outside of a magnet.
Inside the magnet, it goes from south to north.

The magnetic poles of the Earth are notorious
wanderers.
In May 2001, the North
Magnetic Pole was located at 81.3°N 110.8°W,
moving northwest in the Canadian Arctic
at about 40 km per year (more than 1 mm/s).
If the Ross 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 a
great 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 has reversed
its north-south polarity many times
with 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, only
much
more frequently.

(2003-06-13) From the north side, positive means counterclockwise:
How the right-hand rule determines the sign of axial quantities.

The mnemonic device pictured at left is useful
to identify the polarity of the magnetic field
created by a simple electrical circuit:
Arrows drawn at the extremities of the capital letters "N" or "S" indicate the
direction of the current, when you're watching either
the North or South side of the circuit.
This engineering convention is (fortunately) compatible with the
traditional definition of the north pole of a magnet,
as described above.

N (+)

Happily, the above mnemonic trick also describes the way
the Earth rotates
(counterclockwise, if you are above the geographic north pole).
By analogy,
astronomers and
others call north the side of a rotating celestial body
from which it is seen to rotate counterclockwise
(regardless of whatever magnetic field may prevail around such a body).

Mathematically, a rotation is a pseudovector
(whose magnitude is the angular speed w )
directed northward along the axis of rotation.
This states the same convention as the so-called "right hand rule":
The fingers of your right hand show the direction of motion
in a rotation whose vector is in the direction of your thumb.

The prefix pseudo- is often used to denote a quantity which is
so defined that its sign would change if our notions of
"left" and "right" where reversed (the rotation pseudovector is one example).
Such quantities may also be called axial,
in contradistinction to the more familiar polar quantities,
whose definitions do not depend on any particular convention
concerning "space orientation".

The cross-product of two [polar] vectors is axial, so is the cross-product
of two axial vectors. However, the cross-product of a polar vector and
a pseudovector is polar.
For example, the relation
V = W´R
holds, which gives the velocityV [a polar vector]
for a point of a rotating solid,
if W is the [axial]
vector of rotation,
and R 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, but
the wind is caused by differences in pressure which ultimately come from
heat 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 changes
the temperature of land masses faster than it changes the
temperature 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:

Pattern
of Prevailing Winds at the Surface of the Globe

Name

Latitude

Prevailing Direction

North Pole

Variable (high pressure)

Polar Easterlies

Northern Circumpolar

From NE to SW

Northern

Variable (low pressure)

Westerlies

Midnorthern

From SW to NE

Horse Latitudes

Around 30°N

Variable or calm (high pressure)

Northeast Trade Wind

Northern Tropical

From NE to SW

Doldrums

Equatorial

Variable or calm (low pressure)

Southeast Trade Wind

Southern Tropical

From SE to NW

Horse Latitudes

Around 30°S

Variable or calm (high pressure)

Roaring Forties

Midsouthern

From NW to SE

Southern

Variable (low pressure)

Polar Easterlies

Southern Circumpolar

From SE to NW

South Pole

Variable (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 would
have 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 or
anticyclonic meteorological disturbances
(especially in the zones indicated as "variable" in the above table).

A cyclone is a low pressure zone, around which winds
rotate counterclockwise in the northern hemisphere (clockwise in the southern hemisphere).

An anticyclone is a high pressure zone, around which winds
rotate 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 certainly not 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 virtually all French textbooks
quote the generic linear function as y = ax+b.
If the tradition was of French origin, wouldn't the French use it?

In an earlier forum
on this [apparently popular] subject, John H. Conway 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 find any use
before 1850 of the word "slope" itself to denote the tangent of a line's inclination...

Conway "seemed to recall" that Euler (1707-1783) did use m for slope,
which remains unconfirmed.
However, Dr. Sandro Caparrini (University of Torino) found out
that at least one contemporary of Euler did so, since Vincenzo
Riccati (1707-1775) used the notation y = mx+n
as early as 1757, in a reference to
Jakob
Hermann (1678-1733).
(This and other related facts have been reported online
in the excellent historical glossary of Jeff Miller;
look under Slope.)

Eric Weisstein reports
that the use of the symbol m 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 was y = mx+c.
[Treatise on Plane Co-Ordinate Geometry as Applied to
the Straight Line and the Conic Sections
by I. Todhunter, Macmillan (London, UK) 1888].

The preferred notation for the slope-intercept cartesian equation of
a straight line in the plane is
not at all universal, though.
Here's what we have gleaned so far.

y = m x + n

Vincenzo Riccati (1757)
Netherlands, Uruguay

y = m x + c

UK

y = m x + b

US, Canada

y = a x + b

France
Netherlands, Uruguay

y = k x + b

Russia

y = k x + m

Sweden

y = k x + d

Austria

y = p x + q

Netherlands

Please,
let me know
if you are in a position to add to the above table
(and/or confirm or deny any part of it). Thanks.

Acknowledgements :

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

(Bob of Sacramento, CA. 2000-12-04)
What is the mark at 19" 3/16th on a tape measure,
and why does it repeat 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 to
19"^{ 3}/_{16 }).

The diamond marks are also called "black truss" markings, because they correspond
to 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 appear
every 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 is 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 built
by measuring horizontally as many diamonds as there are vertical feet.

The ratio 8/5 = 1.6 is very close to the so-called
Golden Ratio,
which has been used extensively in architecture since antiquity...
The golden ratio is the aspect ratio of a rectangle whose larger side is to the smaller
side what the sum of the two sides is to the larger side.
It is also equal to the diagonal of a regular pentagon of unit side.
Its precise value is
f = (1+Ö5)/2, which is
about 1.618034.

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

Excel
interprets something like 3^3^3 as (3^3)^3 = 27^3 = 19683.
(For some calculators, this expression would
mean 3^(3^3) = 3^27 = 7625597484987.)

This idiosyncrasy of Excel makes the question interesting, as there is a
keystroke cost for the parentheses that force a so-called 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 N_{n}
expressible in n keystrokes or less.

N1

9

N2

99

N3

9E9 has 10 digits.

N4

9E99 has 100 digits.

N5

9^9E9 has 8588182585 digits.

N6

9^9E99 has over 8.588´10^{99} digits.

N7

9^9E999 has over 8.588´10^{999} digits.

N8

9^9E9999 has over 8.588´10^{9999} digits.

N9

9^(9^9E9) has over 0.95424 ´ N5 digits.

N10

9^(9^9E99) has over 0.95424 ´ N6 digits.

N11

9^(9^9E999) has over 0.95424 ´ N7 digits.

N12

9^EXP(9^9E9) has over 0.95424 ´ exp(N5) digits.

N13

9^EXP(9^9E99) has over 0.95424 ´ exp(N6) digits.

N_{n+7}

9^EXP( N_{n} )

N34

9^EXP(9^EXP(9^EXP(9^EXP(9^9E99))))

N35

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

N36

9^EXP(9^EXP(9^EXP(9^EXP(9^9E9999))))

N37

9^EXP(9^EXP(9^EXP(9^EXP(9^(9^9E9)))))

N38

9^EXP(9^EXP(9^EXP(9^EXP(9^(9^9E99)))))

There is a good reason to
stop the table at this point, as explained below.

A nice function to generate large numbers is the factorial
function. However, its Excel name (FACT) is longer than that
of 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 possibly 9^(9^(9^9E9))...
Because the argument of EXP has one more keystroke, the corresponding
expression is the largest by a wide margin. The somewhat larger
growth 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 an inner expression
found 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 in Excel 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 pure Excel idiosyncrasy, not shared by similar languages).
The basic idea is to use compact descriptions of extremely long text
strings (representing syntactically correct numbers) using the function
REPT(x,n),
which returns n concatenated copies of whatever string is specified by x.
Such strings can be explicitly converted into syntactically acceptable numbers
using the EVALUATE function (which works if macros are allowed).
Either one of the following related patterns will thus convert a numerical
expression W of length n into
an expression of length 2n+35 or 2n+36 representing a tower of
W 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 above
for 37 or 38 keystrokes.

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 to
den0eng3
for suggesting the use of EVALUATE...

Largest number N_{n}
expressible in n keystrokes (continued).

N39

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

N40

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

N41

EVALUATE(REPT("9^(",9E9)&9&REPT(")",9E9))

N42

EVALUATE(REPT("9^(",9E9)&99&REPT(")",9E9))

N43

EVALUATE(REPT("9^(",9E99)&9&REPT(")",9E99))

N44

EVALUATE(REPT("9^(",9E99)&99&REPT(")",9E99))

N43

EVALUATE(REPT("9^(",9^9E9)&9&REPT(")",9^9E9))

N44

EVALUATE(REPT("9^(",9^9E9)&99&REPT(")",9^9E9))

N_{2n+35}

EVALUATE(REPT("9^(", N_{n})&9&REPT(")", N_{n}))

N_{2n+36}

EVALUATE(REPT("9^(", N_{n})&99&REPT(")", N_{n}))

Finding what string of given length describes the largest number in a given formal
language is a variant of what's known as the Busy Beaver Problem
(the name comes from the impression you get from watching small Turing machines
generate large outputs).
For a general enough computer language
(i.e., as "powerful" as a lowly Turing machine)
it can be shown that no 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 also unabridged answer.]
What are the odds in favor of being dealt a given poker hand?

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

The probability of a given type of hands is thus the number of such hands
divided by 2598960.
When the probability of something is the fraction P = x / (x + y) ,
its so-called odds are said to be either x to y in favor
or y to x against, as shown in the table below,
which assumes some familiarity with poker
(10 kinds of "straights" are normally allowed,
see below or here).

Type

Number of Hands

Probability

Odds in Favor

Royal Flush

C(4,1) C(1,1)

4

1 / 649740

1 to 649739

Straight Flush

C(4,1) C(10-1,1)

36

3 / 216580

3 to 216577

4 of a Kind

C(13,1) C(48,1)

624

1 / 4165

1 to 4164

Full House

13 C(4,3) 12 C(4,2)

3744

6 / 4165

6 to 4159

Flush

C(4,1) [C(13,5) - 10]

5108

1277/649740

1277 to 648463

Straight

C(10,1) (4^{5}-4)

10200

5 / 1274

5 to 1269

3 of a Kind

13 C(4,3) C(12,2) 4^{2}

54912

88 / 4165

88 to 4077

Two Pairs

C(13,2) C(4,2)^{2 }44

123552

198 / 4165

198 to 3967

Pair

13 C(4,2) C(12,3) 4^{3}

1098240

352 / 833

352 to 481

High Card

(C(13,5)-10) (4^{5}-4)

1302540

1277 / 2548

1277 to 1271

TOTAL

C(52,5)

2598960

1

1 to 0

In the last entry,
"High Card" means a hand that's none of the above :
Two such hands would be compared highest card first to decide who wins.

Note that there are normally 10 different "heights" for a straight and
that the ace (A) belongs to the lowest (A,2,3,4,5) and the highest (10,J,Q,K,A),
which is traditionally called a Royal Flush if all cards belong to the same suit.
Should your own local rules disallow the tenth straight 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 respectively
by 4, 32, 5112, 9180 and 1303560 = (C(13,5)-9)(4^{5}-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 the
date 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, since
the latest date appearing elsewhere on the page can't possibly predate the actual year of
printing by more than a century...
Please, let me know if you have
any information about approximately when 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 be at least
10 000 000.

However, for all 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 the Species 2000 project,
the ALL Species Foundation, or the
Census of Marine Life
(CoML).

Originally, CoML was an initiative of the
Consortium for Oceanographic Research and Education
(CORE)
before its 2007 merger with
the Joint Oceanographic Institutions
(JOI) which resulted in a new
entity dubbed Consortium for Ocean Leadership
(COL, activated in 2008).

In his 1992 book The Diversity of Life (Harvard University Press),
Edward O. Wilson quotes a total number of 1402 900 identified species.
This inventory includes 751 000 insects, 123 400 noninsect arthropods, 106 300
other invertebrates, and only 42 300 vertebrates
(less than 10% of which are mammalian).
The remainder consists of
248 400 plants, 69 000 fungi, 57 700 protists
(inluding 26 900 phototropic algae),
and 4800 bacteria (the bacterial world is almost uncharted,
see below).
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. May
offers 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 in
prehistoric times.
In his 1994 book Vital Dust (BasicBooks),
1974 Nobel laureate Christian de Duve quotes all of
the above and calls this the biological equivalent of the burning of the
library of Alexandria in 641.

Even if cataloguing them would essentially be an endless task,
the number of bacterial species in a given sample can be
estimated
statistically by measuring only the total population and the number of
individuals in the most prominent species.
Dr. Tom Curtis (and his coworkers at the
University of Newcastle upon Tyne)
did just that in a recent article
of the Proceedings
of 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 species
of 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 what
the DNA of a group's common ancestor was like.
They can also determine what species is closest to that ancestor
and is thus the most primitive of them, probably because it
has 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?

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 nominal
value, but practical tolerances were much tighter.

In the context of coins, the troy ounce of 480 grains is more appropriate than
the common avoirdupois ounce of 437.5 grains. So your first answer is that
it takes 10 (pre-1982) US pennies to make a troy ounce
(and about 9.1146 of these to make an avoirdupois ounce).

Because newer pennies are about 20% lighter (nominally 2.5 g, as stated below),
the average mass of a penny in a mix of new and old ones is roughly 10% less than
a pre-1982 penny, and it's therefore close enough to 1/10 of an avoirdupois ounce
(see next article).
This approximation is used [without explanation] in the superbly crafted
MegaPenny Project at kokogiak.com.

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].

Some pennies manufactured in 1982 were reported to have a plating 3 times thinner
(99.2% zinc, 0.8% copper). We are unable to confirm.

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

Let's put ourselves in the shoes of whoever had to design the 1982 penny.
To keep a constant size, the new penny should roughly be to the old one in terms of mass
what zinc (7.1 g/cc) is to copper (8.9 g/cc) in terms of density.
A more precise estimate may be obtained by looking up precise density values.
We found copper listed at 8.960 g/cc at 20°C.
Zinc is listed at 7.133 g/cc at 25°C. To get its density at 20°C,
we use Zinc's coefficient of linear expansion (listed as
3.02´10^{-5}/°C at 25°C)
so its density at 20°C is about
(1 + 4.53´10^{-4 })
larger, or about 7.136 g/cc. We may assume --although that's
not 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 component
metals taken separately: X grams of pure copper (density d1)
on one side and (1-X) grams of pure zinc (density
d2) on the other.
With this assumption, the density of such an alloy is:
1/(X/d1+(1-X)/d2).
This makes the density of old pennies about 8.847 g/cc and that of new ones about
7.171 g/cc, for a ratio of about 81.06%. As the old penny weighs a nominal
48 grains (about 3.1103g), a new penny of the exact same size would weigh
about 2.521 g. At a nominal mass of only 2.5 g, a new penny has therefore a
volume which its about 0.84% smaller than the volume of an old penny...

The volume of a penny is very close to 0.35 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 new
nominal 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 an
avoirdupois pound (7000 grains). There are 200 nominal dimes in 7000 grains.

Isotopic Pennies.

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

Thanks to Gene Nygaard
of Crosby (ND) for pointing out the nominal masses of the
new penny (2.5 g), the nickel (5 g) and other coins...
$20 per pound is the current standard for dimes and quarters
(since 1965) as well as half-dollars (since 1971).
The same was true of the so-called "Eisenhower dollars" minted between 1971 and 1978
(1½" or 38.1 mm in diameter,
350 grains or about 22.68 g in weight).
From 1873 to 1964, on the other hand,
dimes, twenty-cent pieces, quarters and halves were $40 per kg.
In other words,
new coins are to the old ones what an avoirdupois pound (453.59237 g)
is to a metric pound (500 g).
That is, the mass of new coins is nominally 90.718474% that of the old ones.
Between 1965 and 1970, half-dollars were nominally 11.5g, which is about
1.4% higher than the current value of 175 grains (11.33980925 g).
For completeness, we should also state that current US dollar coins
have a nominal mass of 125 grains (about 8.1 g)
yielding 56 dollar coins to the pound...
The newer golden Sacagawea dollar (first minted in 2000)
was designed to have the exact same weight, size, and
electromagnetic signature as the Susan B. Anthony dollar,
so both coins can be used concurrently in vending machines.

The size and mass of US coins has been legally enacted in
31 USC 5112
(Title 31, Section 5112 of the
U.S. Code, also available from
findlaw.com).
As of 2001,
paragraph (a) of Sec. 5112 still gives the specifications of the old pre-1982 penny;
the leeway necessary for the "new" post-1982 penny is provided by paragraph (c).

US coin denominations withdrawn from circulation include the
half-cent
(1793-1857), 2¢ (1864-1873),
3¢ silver (1851-1873), 3¢ nickel (1865-1889),
half-dime (1792-1873),
and 20¢ piece (1875-1878).
There are arguments
for and
against the withdrawal of pennies
(some polls show that 70% of Americans wish to keep them).

US gold coins have been minted in the following denominations:
$1 (1849-1889), $2½ "quarter eagle" (1796-1929),
$3 (1854-1889), $5 "half eagle" (1795-1929),
$10 "eagle" (1795-1933), $20 "double eagle" (1850-1933). Current
law would only allow the minting of $5, $10, $25 and $50 gold coins...

(Robert of Clifton, TX. 2000-11-11)
US Pennies by the pound.
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.

The price of copper had risen to $1.33 per pound in 1980,
so pennies could not be minted for less than their monetary value.
Copper-plated coins postponed the crisis
(in 2000, it cost 0.81¢ to mint a penny).

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 between
146 and 181 pennies, depending on the percentage of pre-1982 pennies in it.
According
to 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) approximately exp(-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 today
as 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 about
162 pennies in a poundas of November 2000.
If there's already more than 160 pennies in a pound,
the average penny is already slightly less than 1/10 of an avoirdupois ounce!

The above theoretical approach tells how the number of pennies in a pound varies with
time, if we assume that the total number of pennies in circulation
is held roughly constant...
However, it's all based on the "approximate life span" of 25 years quoted by the US Mint,
which could be an overestimate for pennies.
If we are to believe some fundraisers,
an average pound of pennies was already worth $1.64 (164 pennies to the pound)
as early as September 1995,
only 13 years after the introduction of the new penny.
164 pennies in a pound (of 453.59237 g) corresponds to a proportion (x)
of pre-1982 pennies which is such that:
164 = 453.59237 / (3.11034768 x + 2.5 (1-x))
This would mean that x was already as low as 43.55% after only 13 years or so,
implying that the average number of years (T)
that a penny lives is such that exp(-13/T) = 0.4355.
T would thus be about 15.64 years, about 2/3 of what the US Mint states
for its other coins.

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 US gallons
(namely Winchester gallons of exactly 231 cubic inches,
which are very different from Imperial gallons)...

We'll consider nickels to be perfect cylinders.
Packing identical solids as densely as possible
is a notoriously 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 other
in 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 highest
number of nickels per gallon in large containers...

The nominal diameter of a nickel is 0.835", or 21.209 mm
(see 31 USC 5112).
The US
Mint online specifications
give the thickness of a nickel as 1.95 mm,
but I did a quick reality check
by measuring a stack of 20 nickels
and found it to be almost exactly 37 mm, instead of the expected 39 mm!
Assuming a one-digit typo in the official site,
we'll thus take the thickness of a nickel to be 1.85 mm.
The discrepancy is otherwise much too large to be attributed to normal wear...
[Similar thickness measurements for other types of circulated coins match
the nominal 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 the same thickness circumscribed to it.
The top surface area is ½Ö3
times the square of the coin's diameter.
As there are exactly 25.4 mm to the inch,
the above numbers make this volume in cubic inches equal to:

V =
½Ö3 (0.835)^{2} (1.85/25.4)
= 0.0439786196844...

Since there are exactly 231 cubic inches in a US gallon, this translates into
231/V, or about 5252.5523 nickels per gallon.
In 5 gallons, you'd have at most 26262 nickels,
worth $1313.10
(possibly a little bit more if the walls of the container
are shaped to fit the coins, but much less in a random packing).

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

Similarly, a gallon would contain [at best] roughly 10500 dimes, 4200 quarters or
2100 half-dollars. Any of these translates into $1050 per gallon.
The result is the same for these 3 types of coins for a reason you have to figure out,
but here's a clue: Any mixture of these three types of coins
represents the same amount of money, by weight.
Seasoned geometricians would remark that a given volume would be less
valuable with several coin types instead of a single one.
All this, of course, is for the ideal densest possible packing...
In practice, YMWV.

I just counted my very, very full to the rim gallon of pennies. I
only found 5612 pennies, instead of your estimate of 7771.

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