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Previous Puzzles of the Month + Solutions
February 2004

 Puzzle #94 Quiz/test #4 W-kammer #4
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Solution
to puzzle #94
Cut the piece of cardboard (as shown below) to make an open box of the largest possible volume. What would be the best value for x to manage this task succesfully?

 Area A of the open box: A = (7 * 5) - 4x2 = 35 - 4x2   A is 'maximal' when x = 0 Volume V of the box: V = (7-2x)(5-2x)x = 4x3-24x2+35x   Thus 0 < x < 2.5 To know where the variable x peaks we take the derivative of V at the point 0: V' = 12x2-48x+35 = 0 Using the quadratic formula we find: x = 2 - 39/6 ~ 0.959167... Supposing that the corners that are to be cut are squares whose sides are x units long, here is the useful formula to find the value x for cutting an open box of the largest possible volume from any rectangular cardboard: x = [(L+W) ± (L2-LW+W2)]/6  If L=W, then the formula is: x = L/6 (L=Length, W=Width of the cardboard)
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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: the incredible Pythagora's theorem December 98: the cunning areas January 99: less is more, a square root problem 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...

Quiz #4
 Test your word knowledges online 1. If you were to spell out numbers, how far would you have to go until you would find the letter 'A'? 2. Find a word containing the letters "zzs" in its middle. 3. Rearrange the letters in the words 'new door' to make one word complete complete complete

 Everyone has at least one logic or math puzzle that is his or her favorite. Send us yours and let all our readers enjoy them!

Posted puzzles
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 Puzzle #3, logic, by Kh. Guili Why is it very common to have a 9 minute snooze interval on alarm clocks and not 10 instead? Rate: •••• Solution #3 Puzzle #4, logic, by Augusto P. 50 ants are dropped on a 2-meter stick. Each one of them is traveling either to the left or to the right with constant speed of 1 meter per 1/2 minute. When 2 ants meet, they bounce off each other and reverse direction. When an ant reaches an end of the stick, it falls off. What is the longest amount of time to wait that the stick has no more ants? Rate: •• Solution #4

Wunderkammer #4
 Puzzling facts

Network Science or the Small-World Phenomen
The 6 degrees of separation hypothesis basically states that any 2 people on Earth are connected by no more than six levels of acquaintances. Even in the vast confusion of the World Wide Web, on the average, one page is only about 16 to 20 clicks away from any other...
Strogatz and Watts offered a mathematical explanation for the results of a landmark experiment performed in the 1960s at Harvard by social psychologist Stanley Milgram. The researcher gave letters to randomly chosen residents of Omaha and asked them to deliver the letters to people in Massachusetts by passing them from one person to another. The average number of steps turned out to be about six!
Following their experiment, Strogatz and Watts created a mathematical model of a network in which each point, or node, is closely connected to many other nodes nearby. When they added just a few random connections ('short-cuts') between a few widely separated nodes, messages could travel from one node to any other much faster than the size of the network would suggest. The six degrees of separation idea works, because in every small group of friends there are a few people who have wider connections, either geographically or across social divisions.
Until recently, we assumed that it would be our close relationships that would bring us the information and opportunities we have been looking for, but the science of networking says different. It is our bare acquaintances, our friends of acquaintances, who can play crucial roles in our lives. These kinds of relationships are called weak or loose ties. The people we hang out with don't often give us the breakthrough contacts or information we want because, generally speaking, we know the same people and the same information that they know.
The 6 degrees of separation can be demonstrated statistically. Assuming that a person only knows 45 people, and each of these know 45-n non-redundant people and so up to six degrees, this multiplies out to be:

 45 = 45 x 44 = 45 x 44 x 43 = 45 x 44 x 43 x 42 = 45 x 44 x 43 x 42 x 41 = 45 x 44 x 43 x 42 x 41 x 40 = 45 1,980 85,140 3,575,880 146,611,080 5,864,443,200 Total 6,014,717,325

Which is enough to cover the world's population!

Here is the math formula to calculate approximately the degree of separation of any population:
d = log N / log k, where N represents the actual number of people; and k, the average number of acquaintances per person. Using the example above, we obtain:
d = log 6,014,717,325 / log 45 = 5.9... which can be rounded to 6!

The small-world phenomenon could provide answers to a wide range of practical questions, such as how ideas spread, how fads catch on, how a small initial failure can cascade throughout a large power grid or financial system, and how companies can foster internal networks to cope with rapidly changing competitive environments.

 Six degrees of separation... Partecipate to the Small-World project of Columbia University. Nexus: Small Worlds and the Theory of Networks
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