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corner top left Previous Puzzles of the Month + Solutions  
February 2004   

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logo puzzle of the month 1 Puzzle #94
Quiz/test #4 logo pzm 2
logo pzm 3 W-kammer #4
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Solution
to puzzle #94
Send us your comments on this puzzle
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? francais/italiano
open box problem

solution
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 - square root39/6 ~ 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) ± square root(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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Quiz #4
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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

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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.

degree of separation
Six degrees of separation...
Partecipate to the Small-World project of Columbia University.
NEXUS
Nexus: Small Worlds and the Theory of Networks
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n factorial approximation
A formula for approximating n factorial

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