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Previous Puzzles of the Month + Solutions

April - May 2006  

thinking man
logo puzzle of the month 1 Puzzle #106
Quiz/test #16 logo pzm 2
logo pzm 3 W-kammer #16
   Enjoy solving Archimedes' Lab™ Puzzles!

triangle-square Puzzle #106

Magic Probabilities...
  There are on the table 3 playing cards face-down. 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?

probable card

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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 'non-King of Spades' 1, and the other one 'non-King 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 'non-King 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 'non-King of Spades' 1. The game host has deliberately shown the other 'non-King of Spades' card 2.
  b) You originally picked the playing card hiding 'non-King of Spades' 2. The game Host has deliberately shown the other 'non-King 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 'non-King 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 'non-King 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".

Previous puzzles of the month...
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circle-triangle Quiz/Test #16 TOP
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Numbers and Remainders
1. A clown agrees to work in a circus on the condition that he is paid 2 schlonks for every day that he works, while he forfeits 3 schlonks every day that he doesn't work. After 30 days, he finds he has paid out exactly as much as he received. How many days did he work? 2. I have an amount of dollars - less than 100! - that repeatedly divided by 3, the remainder is 2, by 5 the remainder is 3; and by 7 the remainder is 2. How many money do you think I have? 3. To end, an easy one! What is the least number which will divide by the nine digits without leaving a remainder?

Wunderkammer #16: Kordemsky TOP
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"Amusement to an observing mind is study."
- Benjamin Disraeli

Our preferred Russian Math Authors (II):
Boris Anastasevich Kordemsky (1907-1999)

matematicheskaya smekalka book  Boris A. Kordemsky (spelled Kordemskij in the Slavonic countries), a Russian popularizer of recreational mathematics, is probably the best-selling 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 post-Stalin 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 (1882-1942), 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 fun-to-solve problems that have kept the best Russian math and physics students biting their pencils as far back as the time of the czars...

arrow Suggest an ORIGINAL Wunderkammer fact

matemagica cover
MATEMAGICA Blackline masters for making over 25 funny math puzzles! (in Italian). Ideal for math workshops.
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ARCHIMEDES is an interactive review devoted to entertaining and involving its readers with puzzles, recreational mathematics and visual creativity.
archimedes review
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Book of the Month
liar paradox...
The Liar Paradox and the Towers of Hanoi: The Ten Greatest Math Puzzles of All Time
Each chapter covers a main puzzle, the historical and philosophical background behind it...
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"We were so poor my daddy unplugged the clocks when we went to bed"
  -- Chris Rock.


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