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corner top left Previous Puzzles of the Month + Solutions  
June-July 2004  

thinking man
logo puzzle of the month 1 Puzzle #98
Quiz/test #8 logo pzm 2
logo pzm 3 W-kammer #8
   Enjoy solving Archimedes' Lab™ Puzzles!

triangle-square-circle Puzzle #98  
Pizza’s pitfalls
  You and a friend of yours wish to share a perfect circular pizza... How would you split it into real EQUAL parts if you don’t know where the center of the pizza is? (PS. You can use only a kitchen knife and a triangle - setsquare, in UK - and you can't fold the pizza!)
pizza puzzle
Solution A ('fair way' method)
To have subjective equal parts, one of you cuts the slices, the other chooses...

Solution B ('triangle' method)
triangle and pizzaThis method works only if the triangle is equal or larger than the diameter of the pizza.
1. Put the right angle of the triangle on perimeter of the pizza.
2. Then the two sides of the right angle will meet the perimeter in two points A and B. Mark these two points with the knife.
3. Using the triangle as a ruler, cut the pizza in a straight line (the line that passes through points A and B) and you'll get 2 equal parts.

Solution C ('ABC' method)
pizza slicing A1. Mark a spot (A) on the edge of the pizza by nicking the crust with the knife.
2. Using the length of the knife as a guide, similarly mark two more spots (B and C) at equal distances on each side of A.
3. Cut from B to C.
4. With the help of the triangle make another cut starting from A, perpendicular to the first cut and continuing to the other side of the pizza. You have then 2 perfect equal parts. That's all...

Solution D
1. Pick an arbitrary point in the pizza (O).
2. With the help of the triangle, cut the pizza into 8 slices by cutting at 45 degree angles through point O, and imagine that alternate pieces are colored in brown and green. Surprisingly, the area of all the brown slices will always be equal the total area of the green slices! So, to get equal parts, each one of you have to pick up slices of the same color... This theorem can be proved by using calculus and polar coordinates.
Question: if the number of slices is 4, and the slices are cut at 90 degree angles through an arbitrary point in the pizza, does this theorem still work?

pizza slicing B

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circle-triangle Quiz #8 TOP
Test: play with words
1. What is broken when you name it? 2. The only 15-letter word that can be spelled without repeating a letter is... 3. Where is the only place that yesterday always follows today?

Wunderkammer #8 TOP

Give me the fruitful error any time,
full of seeds, bursting with its own corrections.
You can keep your sterile truth for yourself!
Vilfredo Pareto

The Pareto Principle
(also known as the '80-20 Rule', the 'law of the vital few' and the 'principle of factor sparsity')

Vilfredo Pareto  The Pareto principle was suggested by management thinker Joseph Juran. It was named after Vilfredo Pareto, a noted Italian economist and sociologist, who made several important contributions to economics, especially in the study of income distribution and in the analysis of individuals' choices.
  The Pareto principle is a mathematical formulation which states that the distribution of incomes and wealth in society is not random, but exhibits a consistent pattern. This relationship follows a regular logarithmic pattern and can be charted in a similar shape, regardless of the time period or country studied.
  The formula is: Log N = log A + m log x
where N is the number of income earners who receive incomes higher than x, and A and m are constants. In simplified terms, 80% of the wealth is owned by 20% of the population. In its generalized form, the principle states that for many phenomena 80% of consequences stem from 20% of the causes.
pareto vs normal distribution  Hereafter are a few examples where the Pareto principle typically applies:
• 80 % of the traffic pollution is produced by 20 % of the vehicles,
• 80% of the traffic travels on 20% of the roads,
• 80 % of a stock is filled with 20 % of the products,
• 20 % of the customers account for 80 % of the sales volume,
• 80 % of the profit is achieved with 20 % of the customers,
• 80% of customer complaints are about the same 20% of projects, products or services,
• 80% of your measurable results and progress will come from just 20% of the items on your daily to-do list,
• 80% of the clothes you wear are from 20% of your closet,
• 20% of Archimedes' Lab web pages are viewed by 80% of our visitors...
  Sure, those figures are but rough approximations. They all emphasize the highly non-linear distribution of causes and effects or of means and objectives.
  Employment of the Pareto principle improves everyday problem-solving efficiency greatly. Rather than wasting time, energies and money on efforts to correct everything, it is more profitable to focus the attention only on those few variables, which are shown to account for most of the problem.

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Puzzle #11, logic, by Zigmund Froid, D
My mother said: "I've placed 10 dollars in your textbook between pages 125 and 126...", "Oh, thanks Mom!" I answered, but most probably the bill will be somewhere else. Why?
Rate: ••• Solution #11

Puzzle #12, maths, by Agon K. Pech
A passenger fell asleep on the Heidi Express panorama train halfway to his destination. He slept till he had half as far to go as he went while he slept. How much of the whole trip panorama has he missed?
Rate: •• Solution #12

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Month's Quote

"The ultimate truth is
penultimately always a falsehood"
Arthur Koestler

Math Gems

T = 21/g = 1/f
Pendulum period
g = gravitation constant
f = pendulum frequency

optical illusion
  The Shadoks are a kind of outer-space bird-like creatures designed and created by Jacques Rouxel in the late sixties... The Shadoks live on the shadok planet where everything is going wrong, and their efforts to escape to Earth are constantly foiled by rivals from another planet. They speak a special language called "gabuzome". Actually, it is composed of only four syllables: ga, bu, zo, me (pr.: gah, büh, zoh, möh)... 

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