Puzzle #98 Quiz/test #8 W-kammer #8
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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!) Solution A ('fair way' method) To have subjective equal parts, one of you cuts the slices, the other chooses... Solution B ('triangle' method) This 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) 1. 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? 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... Back to puzzle-of-the-month page Get the Archimedes Month's puzzles on your web page!

Quiz #8

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? complete complete complete

Wunderkammer #8

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')   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.   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. Suggest an ORIGINAL Wunderkammer fact

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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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