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solving Archimedes' Lab™ Puzzles! 


Puzzle
#106 


Magic
Probabilities...
There are on the table 3 playing cards facedown. You have to guess
which one is a King of Spades. You pointed the card C... OK, card B is turned
over by our game Host and it isn’t a King of Spades. Now, if you change
your mind and select card A instead of C, what are your winning probabilities?




There
are 3 playing cards and two of
them aren't a King of Spades.
Let's call one of the cards that
isn't a King of Spades 'nonKing
of Spades' 1, and the
other one 'nonKing of
Spades' 2.
1. If
we assume:
The game Host knows where
the King of Spade is;
and you know that the game Host
knows where the King of Spades
is... Then, the chance of winning
is doubled when you switch to the
other playing card A rather than
sticking with your original choice
(card C), because the game Host
turned deliberately a 'nonKing
of Spades' card over.
At
the point you are asked whether
to switch there are 3
possible situations corresponding
to your initial choice, each
with equal probability (1/3):
a) You originally picked the playing card hiding
'nonKing of Spades' 1. The game host has deliberately shown the other 'nonKing
of Spades' card 2.
b) You originally picked the playing card hiding
'nonKing of Spades' 2. The game Host has deliberately shown the other 'nonKing
of Spades' card 1.
c) You originally picked the playing card hiding
the King of Spades. The game Host has shown either of the two 'nonKing of Spades'
cards.
If
you choose to switch, you win
in the first two cases (a and b).
A player choosing to stay with
the initial choice wins in only
the third case (c).
Since in 2 out of 3 equally likely
cases switching wins, the odds
of winning by switching are 2/3.
In other words, a player who
has a policy of always switching
will win on average two times
out of the three.
2. If
we assume:
The game Host doesn’t
really know where the King of Spades
is; and you know that
the game Host doesn’t know
where the King of Spades is...
Then, when the game Host turns
the 'nonKing of Spades' card over,
the probability that you originally
picked the King of Spades increases
from 1/3 to 1/2. The odds are in
this case equal, wether you switch
your initial choice or not.
3. If
we assume:
You doesn’t really
know whether the game Host knows
where the King of Spades is or
not.
The probability n that
you originally picked the King
of Spades is then: 1/3 < n < 1/2.
Switching in this case is still
advantageous, your odd ratios being
ranged between 1/2 and 2/3.
The
Paradox of Bertrand's Box
A similar probability puzzle was
published by the mathematician
Joseph Bertrand (no relation to
Bertrand Russell) in his 1889 text
'Calcul des Probabilités'.
The reader is asked to imagine
three desks, with two drawers each.
He knows that one desk contains
a gold medal in each drawer, one
contains a silver medal in each
drawer, and one contains one of
each. He doesn't know which desk
is which. The question posed was:
if the reader opens a drawer and
discovers a gold medal, what are
the chances that the other drawer
on that desk also contains gold?
A
well known adaptation of this
puzzle is the "Monty Hall
problem".


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Wunderkammer
#16: Kordemsky 



"Amusement
to an observing mind is study."
 Benjamin Disraeli
Our
preferred Russian Math Authors (II):
Boris Anastasevich Kordemsky (19071999)
Boris
A. Kordemsky (spelled Kordemskij in
the Slavonic countries), a Russian
popularizer of recreational mathematics,
is probably the bestselling author
of puzzle books in the history of
the world. His book, Matematicheskaya
Smekalka (The
Moscow Puzzles), sold
more than a million copies in the
former USSR / Russia alone, and it
has been translated into many languages.
B. Kordemsky was a talented high school mathematics teacher in Moscow.
For many years he wrote for Mathematics in School magazine, in which his column “Entertaining
Page” was very popular among both teachers and students. His math and puzzle
columns also appeared in Science and Life, Quantum, and Young
Technician.
It is for his book The
Moscow Puzzles, however, that Boris
was most known. It was first published in 1954, just after the postStalin thaw
in the Soviet Union, in an edition of 150,000 copies, which quickly sold out.
A second edition in 1955 sold a similar number, as did a third edition in 1956.
Altogether, more than 10 editions have appeared in Russia to date. The book has
been translated into Ukrainian, Bulgarian, Romanian, Hungarian, Czech, Polish,
German, French, Italian, Chinese, Japanese, Korean, all the Baltic languages,
English, and others...
The
Moscow Puzzles is a huge collection of math puzzles of many
kinds: dissections, tessellations, magic squares, cryptarithms, puzzles with
dice and dominoes, matchstick puzzles, algebraic problems, and assorted brainteasers.
Gardner called it “marvelously varied”. Math puzzle enthusiasts may
well recognize some of the puzzles featured in the book as the original inventions
of the Italian Luca Pacioli, the American Sam Loyd, the British Henry Dudeney,
the Belgian Maurice Kraitchik, and others. But undoubtedly a significant number
of puzzles were new (or readapted in an interesting way).
The only other Russian author who can be compared with Kordemsky
is Yakov
Perelman (18821942), who in addition to books on recreational maths,
wrote similar books on mechanics, physics and astronomy.
Related Books 

The
Moskow puzzles
Boris
A. Kordemsky
Most popular Russian puzzle book
ever published. Marvelously varied
puzzles ranging from simple "catch" riddles
to difficult problems. Lavishly
illustrated with clear diagrams
and amusing sketches.


The
Chicken from Minsk
Yuri
Cherniak, Robert Rose
The 99 brainy and infuriatingly
funtosolve problems that have
kept the best Russian math and
physics students biting their
pencils as far back as the time
of the czars...


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ORIGINAL Wunderkammer fact 





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