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Previous Puzzles of the Month + Solutions

August - September 2006  

thinking man
logo puzzle of the month 1 Puzzle #108
Quiz/test #18 logo pzm 2
logo pzm 3 W-kammer #18
   Enjoy solving Archimedes' Lab™ Puzzles!

triangle-square Puzzle #108 TOP

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)?


show-hide Click to show/hide

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)?

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...
puzzle solver
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circle-triangle Quiz/Test #18 TOP
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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?


Wunderkammer #18: Medieval Mickey TOP
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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

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?

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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Did you enjoy our puzzles and our optical illusions? You can find them every month in FOCUS Giochi magazine!

Focus Giochi

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Book of the Month

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"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:
math relationship
an interesting relationship between an infinite sum and a product over all primes


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