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# Previous Puzzles of the Month + Solutions

August - September 2006

 Puzzle #108 Quiz/test #18 W-kammer #18
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 Puzzle #108

Double-pan precision balance
Weighing from 1 to 60 grams using weights on one side of a balance is possible with weights (calibration masses) of 1, 2, 4, 8, 16 and 32 grams. But is this possible with another sextuplet of weights? Or, better still, by using a lesser number of weights (for example, 4 weights instead of 6)?

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 If both sides of the balance can be used, then five weights from the powers of 3 will suffice (1, 3, 9, 27, 81) to measure all the weights up to 121 grams: 1, 3-1, 3, 3+1, 9-(3+1), 9-3, 9+1-3, 9-1, 9, 9+1, 9+3-1, 9+3, 9+3+1, 27-(9+3+1), etc. However, for measuring weights up to 60 grams there are also many other alternative solutions, e.g.: 1, 3, 9, 20, 27, or even 1, 2, 7, 19, 31... etc. Why it isn't possible to determine weights from 1 to 60 using only four weights (calibration masses)? Because: A) If weights are on one side only, then each weight can have 1 of 2 states: 1) 'on the scale' or 2) 'off'. With only five weights, only 32 cases can be achieved (including 'all off' = 0). Therefore, 6 weights at least are required for measuring weights up to 60 grams. B) If weights are on both sides, each weight can have 1 of 3 states: 1) 'on left side', 2) 'on right side', or 3) 'off'. With only four weights, 34 or 81 states can be achieved. One is 'all off' = 0. The other 80 comprise 40 pairs where the positions of the weights are mirrored. Thus, four weights can give only 40 unique overall weights... But there is a logical way to determine weights from 1 to 60 using only four weights! When considering weights from 1 - 60 grams, to measure 60 discreet weights, you need only to verify 30 discreet weights. From 1 - 100 grams, you would need 50 not 100 etc. The logic is as follows: - NOT 0, NOT 2, then 1 - NOT 2, NOT 4, then 3 - NOT 4, NOT 6, then 5 ... etc. As you can see you need only to verify even weights. For 1-60 grams you need only four weights (calibration masses). If both sides of the balance can be used, you can use the weights 2 grams, 6 grams, 18 grams, and 54 grams as follows: 2, 6-2, 6, 6+2, 18-(6+2), 18-6, (18+2)-6, 18-2, 18, 18+2, (18+6)-2, 18+6, 18+6+2, 54-(18+6+2), 54-(18+6), ... , 54+6. The winners of the puzzle of the month are: Gord Steadman and Larry Bickford. Congratulations!

Previous puzzles of the month...
 August 98: the irritating 9-piece puzzle September 98: the impossible squarings October 98: the multi-purpose hexagon November 98: Pythagora's theorem December 98: the cunning areas January 99: less is more (square roots) February 99: another square root problem... March 99: permutation problem... April 99: minimal dissections July 99: jigsaw puzzle August 99: logic? Schmlogic... September 99: hexagon to disc... Oct-Nov 99: curved shapes to square... Dec-Jan 00: rhombus puzzle... February 00: Cheeta tessellating puzzle... March 00: triangular differences... Apr-May 00: 3 smart discs in 1... July 00: Funny tetrahedrons... August 00: Drawned by numbers... September 00: Leonardo's puzzle... Oct-Nov 00: Syntemachion puzzle... Dec-Jan 01: how many squares... February 01: some path problems... March 01: 4D diagonal... April 01: visual proof... May 01: question of reflection... June 01: slice the square cake... July 01: every dog has 3 tails... Aug 01: closed or open... Sept 01: a cup of T... Oct 01: crank calculator... Nov 01: binary art... Dec 01-Jan 02: egyptian architecture... Feb 02: true or false... March 02: enigmatic solids... Apr 02: just numbers... May 02: labyrinthine ways... June 02: rectangle to cross... July-Aug 02: shaved or not... Sept 02: Kangaroo cutting... Oct 02: Improbable solid... Dec-Jan 03: Hands-on geometry Feb-Mar 03: Elementary my dear... Apr-May 03: Granitic thoughts June-July 03: Bagels... September 03: Larger perimeter... Oct-Nov 2003: square vs rectangle Dec-Jan 04: curvilinear shape... February 04: a special box March 04: magic 4 T's... April O4: inscribed rectangle May 04: Pacioli puzzle... June 04: pizza's pitfalls October 04: Odd triangles February 05: Same pieces May-June 05: stairs to square July-August 05: cheese! Sept-Oct 05: magic star Dec-Jan 06: red monad Feb-March 06: cows and chickens Apr-May 06: intriguing probabilities June-July 06: squared strip
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 Quiz/Test #18
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 Math Challenges 1. Find the value of the digit C in the following calculation: AB - BA = C3 (Each letter represents a digit) 2. You can see that: 22 - 1 = 3 is prime. How many other examples of a prime which is one less than a perfect square exist? 3. In a bar, one-half drank only wine, one-third drank only colas, and five people were bar workers. How many people were in the bar? complete complete complete

 Wunderkammer #18: Medieval Mickey
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 "I only hope that we don't lose sight of one thing - that it was all started by a mouse" - Walt Disney Medieval Mickey   An amazing 700-year-old picture that looks just like Mickey Mouse (see above) has been discovered on a church fresco in the village of Malta in the province of Carinthia (Austria).   Walt Disney first sketched the cartoon character in 1928 but an Austrian art historian spotted this uncannily similar drawing.   But represents the picture really a mouse? Art expert Eduard Mahlknecht reckons that the picture was painted around the year 1300, making him 600 years older than the Disney character. He added that he wasn't totally sure the painting was a mouse. Mahlknecht reckons that it could be a weasel because there is a legend all about a weasel that has really big ears.   Siggi Neuschitzer, manager of the local tourism office, confirmed that the legal process to claim the copyright had already started. He affirmed: "I visited Vienna and had a long meeting with our legal team. They have been instructed to demand Disney return the mouse to its rightful home here in Austria... Anyone who has seen our fresco can see it proves that Mickey Mouse is a true Austrian - and was not from Hollywood". Are Austrians kidding us? Suggest an ORIGINAL Wunderkammer fact

 MATEMAGICA Blackline masters for making over 25 funny math puzzles! (in Italian). Ideal for math workshops. More info...
 Did you enjoy our puzzles and our optical illusions? You can find them every month in FOCUS Giochi magazine! More info...
 Book of the Month More books...
 ••• Smile! "I had lunch with a chess champion the other day. I knew he was a chess champion because it took him 20 minutes to pass the salt". - Eric Sykes ••• Math Gems for math geeks: an interesting relationship between an infinite sum and a product over all primes •••
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