Probabilities in the Game of Monopoly® Probabilities in the Game of Monopoly® Table of Contents I recently saw an article in Scientific American (the April 1996 issue with additional information in the August 1996 and April 1997 issues) that discussed the probabilities of landing on the various squares in the game of Monopoly®. I was intrigued enough with this problem that I started working on trying to find the probabilities for landing on the different squares with all of the rules taken into account. I first wrote a C program that simulates a single person rolling the dice and moving around the board a great number of times. I discovered that it is really necessary to model two different strategies. In the process of figuring all of this out I ran into an interesting difficulty. Just as I finished putting together this web page, I realized that there is a more efficient way of making the probability calculations. Long Term Probabilities for Ending Up on Each of the Squares in Monopoly® Back to my homepage.
- StumbleUpon The length of the polygonal spiral is found by noting that the ratio of inradius to circumradius of a regular polygon of sides is The total length of the spiral for an -gon with side length is therefore Consider the solid region obtained by filling in subsequent triangles which the spiral encloses. -gons of side length , is The shaded triangular polygonal spiral is a rep-4-tile. Picture Gallery 97 Funny Videos, Free Games, & Funny Pictures Sign up | Login Picture Gallery 97 [+] 35 of the best pictures submitted throughout the week! [–] 35 of the best pictures submitted throughout the week! Tags: Gallery, Picture halloween costume win dented car awkward engagement epic storm threat win epic intersection well trained puppy product fail awkward headline rahhh!! mobile bar cat photobombs cat hands free awkward wasp nest futuristic car is futuristic practicing ...? delivering milk epic billboard bubble bed prize fail cleavage fail parking lot surprise epic treehouse comeback win big bike little guy parenting fail headshot at first i was like O_O but then i was like -_- 'huge' tree your mom is a liar impending fail ut oh ur anus is asking what? epic spider Daily Gallery: Tues Sept 18, 20... Picture Gallery 100 Daily Gallery: Fri Nov 16, 2012 Daily Gallery: Tues Jan 29, 201... Daily Gallery: Wed Oct 17, 2012 Picture Gallery 99 Daily Gallery: Wed Nov 14, 2012 Daily Gallery: Mon Nov 19, 2012 Daily Gallery: Mon April 2, 201... | Reply
Fibonacci Number The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation with . As a result of the definition (1), it is conventional to define The Fibonacci numbers for , 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ... Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials Fibonacci numbers are implemented in the Wolfram Language as Fibonacci[n]. The Fibonacci numbers are also a Lucas sequence , and are companions to the Lucas numbers (which satisfy the same recurrence equation). The above cartoon (Amend 2005) shows an unconventional sports application of the Fibonacci numbers (left two panels). A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (OEIS A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Saunière in D. The plot above shows the first 511 terms of the Fibonacci sequence represented in binary, revealing an interesting pattern of hollow and filled triangles (Pegg 2003). ends in zeros. and . as
Nerd Paradise : Calculating Base 10 Logarithms in Your Head Calculating base 10 logarithms in your head on the fly is a lot easier than you may think. It is simply a matter of memorization and a little estimation... First memorize all the single digit base 10 logs. Don't worry, it's not as painful as it sounds. I even made the chart for you: Remember this rule from high school? And what about this one, you remember it too? Good. Example #1: base 10 log of 400 That's the same thing as log(4*100) which equals log 4 + log 100. log of 4 you know from the table above. Now you may ask, what if it isn't just a number with a bunch of 0's after it? Example #2: base 10 log of 35 Suppose you wanted to find the logarithm of 35. Our guess: 1.545 Calculator says: 1.544068... Now you can convince all your friends and teachers that you are autistic. Example #3: base 10 log of 290438572: This is fairly close to log(2.9 * 100000000) = log 2.9 + log 108 2.9 is close to 3. Our Guess: 8 + .45 = 8.45 Calculated Answer: 8.46305... User Comments: 14 And so it goes. To recap: Fixed.
Weierstrass functions Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere". Here is an example of one: It is not hard to show that this series converges for all x. In fact, it is absolutely convergent. It is also an example of a fourier series, a very important and fun type of series. It can be shown that the function is continuous everywhere, yet is differentiable at no values of x. Here's a graph of the function. You can see it's pretty bumpy. Below is an animation, zooming into the graph at x=1. Wikipedia and MathWorld both have informative entries on Weierstrass functions. back to Dr.
Folding Paper in Half Twelve Times Folding Paper in Half 12 Times: The story of an impossible challenge solved at the Historical Society office Alice laughed: "There's no use trying," she said; "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen. Through the Looking Glass by L. The long standing challenge was that a single piece of paper, no matter the size, cannot be folded in half more than 7 or 8 times. The most significant part of Britney's work is actually not the geometric progression of a folding sequence but rather the detailed analysis to find why geometric sequences have practical limits that prevent them from expanding. Her book provides the size of paper needed to fold paper and gold 16 times using different folding techniques. Britney Gallivan has solved the Paper Folding Problem. In April of 2005 Britney's accomplishment was mentioned on the prime time CBS television show Numb3rs. The task was commonly known to be impossible. The price is $16.00 including shipping.
- StumbleUpon Global Warming Uncovers Corpses Frozen in Time Photo via Last Days of the Incas Five hundred years ago, three Inca children were left to freeze high in the cold Argentinian Andes as a religious sacrifice. In time, their bodies mummified, having been swallowed in snow and entombed within the glacier, lost to time. When the three Inca children were discovered thanks to melting in the Andes, their well-preserved, mummified remains helped advance archeological knowledge of their rather mysterious civilization. For example, the frozen body of 24-year-old pilot, Benjamin Rafael Pabón, was discovered by hikers in Peru -- over 20 years after his plane crashed in the Andes. "It took me a very long time to acknowledge he might be dead," said the pilot's mother. Photo: Noah Friedman-Rudovsky for The New York Times A recent report from The New York Times sheds light on several fascinating discoveries that have been made amid the melting ice of some of the world's most threatened snow packs.
6174 (number) 6174 is known as Kaprekar's constant[1][2][3] after the Indian mathematician D. R. Kaprekar. This number is notable for the following property: Take any four-digit number, using at least two different digits. 9990 – 0999 = 8991 (rather than 999 – 999 = 0) 9831 reaches 6174 after 7 iterations: 8820 – 0288 = 8532 (rather than 882 – 288 = 594) 8774, 8477, 8747, 7748, 7487, 7847, 7784, 4877, 4787, and 4778 reach 6174 after 4 iterations: Note that in each iteration of Kaprekar's routine, the two numbers being subtracted one from the other have the same digit sum and hence the same remainder modulo 9. Sequence of Kaprekar transformations ending in 6174 Sequence of three digit Kaprekar transformations ending in 495 Kaprekar number Bowley, Rover. "6174 is Kaprekar's Constant".
Nerd Paradise : Divisibility Rules for Arbitrary Divisors It's rather obvious when a number is divisible by 2 or 5, and some of you probably know how to tell if a number is divisible by 3, but it is possible to figure out the division 'rule' for any number. Here are the rules for 2 through 11... The last digit is divisible by 2. The sum of all the digits in the number is divisible by 3. The last 2 digits are divisible by 4. The last digit is 5 or 0. The number is both divisible by 2 and divisible by 3. Cut the number into 2 parts: the last digit and everything else before that. The last 3 digits are divisible by 8 The sum of all the digits in the number is divisible by 9. The last digit is a 0. Break the number into two parts (like you did for the division by 7 rule). Also there is a quick way for determining divisibility by 11 for 3-digit numbers: If the inner digit is larger than the two outer digits, then it is divisible by 11 if the inner digit is the sum of the two outer digits. Rules for all divisors ending in 1... User Comments: 9 Dividing By 12
The Infinity Room With this immersive installation, French artist Serge Salat invites visitors to take a journey through endless layers of space, decked out with cubic shapes, panels of mirrors, shifting lights and music. “Beyond Infinity” is a multi-sensory, multimedia experience that blends Eastern Chinese with Western Renaissance. Inspired by the Suzhou Gardens, a masterpiece of Chinese landscape, the three-lined trigram of I Ching is the main pattern that organizes the space of the work. Salat uses mirrors as optical illusions, exploding a single room into spatial infinity. via [Architizer] Views: 422998 Tags: Serge Salat, The Infinity Room, architecture, design
How To Analyze Data Using the Average The average is a simple term with several meanings. The type of average to use depends on whether you’re adding, multiplying, grouping or dividing work among the items in your set. Quick quiz: You drove to work at 30 mph, and drove back at 60 mph. What was your average speed? Hint: It’s not 45 mph, and it doesn’t matter how far your commute is. But what does it mean? Let’s step back a bit: what is the “average” all about? To most of us, it’s “the number in the middle” or a number that is “balanced”. The average is the value that can replace every existing item, and have the same result. One goal of the average is to understand a data set by getting a “representative” sample. The Arithmetic Mean The arithmetic mean is the most common type of average: Let’s say you weigh 150 lbs, and are in an elevator with a 100lb kid and 350lb walrus. The real question is “If you replaced this merry group with 3 identical people and want the same load in the elevator, what should each clone weigh?” Pros: Cons:
New Mayan calendar discovered: world won't end in 2012 Earth has a new reason to celebrate. It's looking like we will make it past Dec. 21, 2012. According to LiveScience, researchers have unearthed the oldest-known version of the ancient Maya calendar in the Guatemalan rainforest. Archaeologist David Stuart of the University of Texas, who worked to decipher the glyphs, told LiveScience the calendar does not mark the end of the world. In fact, quite the opposite. According to SFGate the calendar, which is said to be exquisitely preserved, was found in a 1,000-year-old house in Guatemala. More from GlobalPost: Mexico uses Mayan doomsday prediction to lure tourists The newly discovered astronomical tables are at least 500 years older than those preserved in the Maya codices, said Science magazine. The ninth-century structure was first found in 2010, according to SFGate, by Max Chamberlain, a student of Saturno. Saturno was extremely excited about their find.