From Newsgroup: rec.puzzles


Time for an oldie but goodie.

You are presented with 14 bronze pennies and informed that exactly one is counterfeit. You have a balance scale and a known good coin and know that
the scale will balance when good coins are compared, but will
dip left or right when a counterfeit coin is compared. (The counterfeit
coin may be either heavier or lighter than a genuine coin.)

What is the minimum total number of weighings required to guarantee
that you can identify the counterfeit coin?


Cheers, etc.
noUb-Anoab!+noe
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From Newsgroup: rec.puzzles

On Thu, 02 Apr 2026 05:24:45 GMT, James Dow Allen wrote:

Time for an oldie but goodie.

Can anyone do better? I suspect there is a more sophisticated solution.

My effort:

V pna qrgrezvar gur pbhagresrvg va sbhe jrvtuvatf, ohg V'z abg fher gung
vf bcgvzny. Pna nalbar qb vg va guerr?

Ynory gur pbvaf: N, O, P, Q, R, S, T, U, V, W, X, Y, Z, A naq jrvtu:

1. {NOPQ}:{RSTU}
2. {NOPQ}:{VWXY}

Ng guvf cbvag jr rvgure xabj N, O, P, Q, R, S, T, U, V, W, X naq Y ner nyy tbbq naq Z naq A ner fhfcrpg, be juvpu tebhc bs sbhe pbagnva gur fhfcrpg
pbva. Vs jr xabj juvpu tebhc bs sbhe vf fhfcrpg, gura jr xabj nyy gra
pbvaf, abg cneg bs guvf tebhcvat, ner tbbq.

Jr pna erfbyir guvf va n znkvzhz bs gjb zber jrvtuvat ol fcyvggvat gur
tebhc bs sbhe fhfcrpg pbvaf vagb gjb tebhcf bs gjb. Jrvtu

3. {FF}:{TT}

Juvpu rire cnve pbagnvaf gur fhfcrpg erfbyir gung jvgu n sbhegu jrvtuvat ntnvafg n xabja tbbq pbva.

4. {F}:{T}
--
David Entwistle
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From Newsgroup: rec.puzzles

On Thu, 02 Apr 2026 05:24:45 GMT, James Dow Allen wrote:

Time for an oldie but goodie.

You are presented with 14 bronze pennies and informed that exactly one
is counterfeit. You have a balance scale and a known good coin and know
that the scale will balance when good coins are compared, but will dip
left or right when a counterfeit coin is compared. (The counterfeit
coin may be either heavier or lighter than a genuine coin.)

Also, given the original problem and no constraint on approach, nor
equipment, what would you propose? Oil filled tank, vibrating plate, air up-draft, what would be your best shot at finding that counterfeit coin
most efficiently?
--
David Entwistle
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From Newsgroup: rec.puzzles


David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> posted:

On Thu, 02 Apr 2026 05:24:45 GMT, James Dow Allen wrote:

Time for an oldie but goodie.

Can anyone do better? I suspect there is a more sophisticated solution.

My effort:

V pna qrgrezvar gur pbhagresrvg va sbhe jrvtuvatf, ohg V'z abg fher gung
vf bcgvzny. Pna nalbar qb vg va guerr?

Yes. This is a very famous puzzle, usually defined with 12 coins but
Fbyhgvba vf cbffvoyr jvgu guvegrra pbvaf, be rira sbhegrra!

Abgr gung {guerr}^x pnfrf pna or qvfgvathvfurq jvgu x jrvtuvatf,
fb guerr-gb-gur-guerr (v.r. gjragl-frira) pnfrf pna or qvfgvathvfurq jvgu
guerr jrvtuvatf vs lbh qb guvf CRESRPGYL.

Gurer nccrne gb or gjragl-rvtug pnfrf (14 cbffvoyr pbhagresrvgf, gvzrf
gjb sbe gur n urnil-yvtug ovg juvpu lbh jvyy yrnea jurgure lbh jnag gb be abg). Ohg guvf erqhprf gb n gjragl-frira zvavzhz! (Ubj?)

Vg jvyy or n IREL gvtug fdhrrmr; znxr fher gung rnpu jrvtuvat qvivqrf gur erznvavat pnfrf RKNPGYL 33%, 33%, 33%.

-- Wnzrf
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From Newsgroup: rec.puzzles


David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> posted:

On Thu, 02 Apr 2026 05:24:45 GMT, James Dow Allen wrote:

Time for an oldie but goodie.

You are presented with 14 bronze pennies and informed that exactly one
is counterfeit. You have a balance scale and a known good coin and know that the scale will balance when good coins are compared, but will dip
left or right when a counterfeit coin is compared. (The counterfeit
coin may be either heavier or lighter than a genuine coin.)

Also, given the original problem and no constraint on approach, nor equipment, what would you propose? Oil filled tank, vibrating plate, air up-draft, what would be your best shot at finding that counterfeit coin
most efficiently?

A related problem, where *almost all the Google search results are WRONG*,
is HOW did Archimedes determine whether the crown of his King was pure gold?

(Naq jub jnf vg perqvgrq jvgu svefg qrqhpvat Nepuvzrqrf' npghny zrgubq?)
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From Newsgroup: rec.puzzles


James Dow Allen <user4353@newsgrouper.org.invalid> posted:

I need to be more careful and provide absolute clarity. Hence some
Addenda:

David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> posted:

On Thu, 02 Apr 2026 05:24:45 GMT, James Dow Allen wrote:

You are presented with 14 bronze pennies and informed that exactly one
is counterfeit.

This should have been in bold-face. EXACTLY ONE of the 14 coins is either heavier or lighter than the other 13 coins.

A related problem, where *almost all the Google search results are WRONG*,
is HOW did Archimedes determine whether the crown of his King was pure gold?

This is the famous story where that great ancient mathematician/physicist
had an inspiration while bathing and allegedly ran naked through the
streets of Syracuse shouting "Eureka!"

I should have emphasized that pure gold was the densest substance known
to the ancients: Any adulterated crown would have a density less
than that of pure gold.

This is no longer true today. In fact Rand Paul and other nuts have suggested that Fort Knox gold has been replaced with plated tungsten.

Tungsten was first isolated as a metal in 1783 by Spanish brothers
Juan Jos|- and Fausto Elhuyar, who reduced it from wolframite.

-- James
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From Newsgroup: rec.puzzles

On Mon, 13 Apr 2026 10:19:34 GMT, James Dow Allen wrote:

Yes. This is a very famous puzzle, usually defined with 12 coins

Interesting. I had to go right back to four coins and two weighings to get
the idea. Well worth the effort during another rainy day.

It has stopped raining now...
--
David Entwistle
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From Newsgroup: rec.puzzles


Is this newsgroup moribund? :-{
What newsgroup, if any, should the following be posted in?

One of my hobbies is reviewing the histories of math and physics.
The story of Archimedes testing his king's crown has an amazing twist.
I hoped that rec.puzzlers would enjoy the amazement! Or share their
knowledge of the solution.

Please, anybody(!), some sort of response?

James Dow Allen <user4353@newsgrouper.org.invalid> posted:

A related problem, where *almost all the Google search results are WRONG*, is HOW did Archimedes determine whether the crown of his King was pure gold?

This is the famous story where that great ancient mathematician/physicist
had an inspiration while bathing and allegedly ran naked through the
streets of Syracuse shouting "Eureka!"

I should have emphasized that pure gold was the densest substance known
to the ancients: Any adulterated crown would have a density less
than that of pure gold.

Much of the story is likely to be apocryphal. But ancient legends often have
a basis in fact, so I'll assume Archimedes DID devise a method for determining the purity of a gold object, even if
"running naked through the streets of Syracuse" was a fiction.

The usual solution originates with Marcus Vitruvius, who lived two centuries after Archimedes. (Vitruvius is one of the 12 people on Girolamo Cardano's list of the 12 greatest geniuses.) This solution is well-known; you can
find it by entering, for example,
archimedes gold crown eureka -ai
into Google Search. (Note that the Gemini "AI" is disabled by the "-ai.")

I performed this search just now and got links to three text files
followed by several YouTubes. All three of the text files had the WRONG answer.
One of the links pointed to a reddit discussion with more than a dozen posts; NONE of them hinted at the correct answer!

It was apparently Galileo Galilei who first published the "correct" solution. An AI told me that afore-mentioned Cardano had published earlier, but
then admitted that that claim was unsourced and probably untrue.

I've been careful not to "spoil" the clever solution to testing the crown.
*Kudos to all who figure it out themselves!!
The "correct" answer (Galileo's solution) has three advantages over the
usual answer given by dozens of books and hundreds of Google hits:

(1) Galileo's solution is EASIER to perform than the "wrong" Vitruvius solution.
(2) Galileo's solution is much more ACCURATE, easily detecting small differences
in density.
(3) Archimedes was one of the greatest of all geniuses and is noted for Archimedes Principle of hydrostatics. The idea that he would have been proud to invent Vitruvius' trivial solution is absurd.

Cheers,
James

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