What is it? Why is it so valuable? Should I buy some? How do I buy some? Yes, this is actually happening! And why not? Imagine a gigantic piece of paper that lists every transaction ever completed.

Creative Exercises Ramanujan's taxi. Ramanujan was an Indian mathematician who became famous for his intuition for numbers. When the English mathematician G. Hardy came to visit him in the hospital one day, Hardy remarked that the number of his taxi was , a rather dull number. To which Ramanujan replied, "No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways. Use four nested for loops.

Now, the license plate seems like a rather dull number. Determine why it's not. The checksum digit d 1 can be any value from 0 to the ISBN convention is to use the value X to denote Write a program ISBN. It's ok if you don't print any leading 0s. Exponential function. Assume that x is a positive variable of type double. Write a program Exp. Write two programs Sin. In the game show Let's Make a Deal , a contestant is presented with three doors.

Behind one door is a valuable prize, behind the other two are gag gifts. After the contestant chooses a door, the host opens up one of the other two doors never revealing the prize, of course. The contestant is then given the opportunity to switch to the other unopened door. Should the contestant do so? Intuitively, it might seem that the contestant's initial choice door and the other unopened door are equally likely to contain the prize, so there would be no incentive to switch.

Write a program MonteHall. Your program should take an integer command-line argument n , play the game n times using each of the two strategies switch or don't switch and print the chance of success for each strategy. Or you can play the game here. Euler's sum-of-powers conjecture. Write a program Euler. Use the long data type. Web Exercises Write a program RollDie. Write a program that takes three integer command-line arguments a, b, and c and print the number of distinct values 1, 2, or 3 among a, b, and c.

Write a program that takes five integer command-line arguments and prints the median the third largest one. How can I create in an infinite loop with a for loop? Solution : for ;; is the same as while true. What's wrong with the following loop? By defining it inside the loop, a new variable sum is initialized to 0 each time through the loop; also it is not even accessible outside the loop.

Write a program Hurricane. Below is a table of the wind speeds according to the Saffir-Simpson scale. Category Wind Speed mph 1 74 - 95 2 96 - 3 - 4 - 5 and above What is wrong with the following code fragment? A better solution is to write if isPositive. There are two different solutions. Boys and girls. A couple beginning a family decides to keep having children until they have at least one of either sex.

Estimate the average number of children they will have via simulation. Also estimate the most common outcome record the frequency counts for 2, 3, and 4 children, and also for 5 and above. What does the following program do? Repeat the previous question, but assume the couple keeps having children until they have another child which is of the same sex as the first child. But the most likely value is 2 for all values of p. Write a program using a loop and four conditionals to print 12 midnight 1am 2am Bob tosses another fair coin until he sees a head followed by a tail.

Write a program to estimate the probability that Alice will make fewer tosses than Bob? Rewrite DayOfWeek. Use a switch statement. Write a program to read in a command line integer between ,, and ,, and print the English equivalent. Here is an exhaustive list of words that your program should use: negative, zero, one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, hundred, thousand, million.

Don't use hundred, when you can use thousand, e. Gymnastics judging. A gymnast's score is determined by a panel of 6 judges who each decide a score between 0. The final score is determined by discarding the high and low scores, and averaging the remaining 4.

Write a program GymnasticsScorer. Quarterback rating. Write a program QuarterbackRating. Use your program to compute Steve Young's record-setting season As of , the all-time single-season record is Decimal expansion of rational numbers. We use the notation 0.

Write a program DecimalExpansion. Hint : use Floyd's rule. Friday the 13th. What is the maximum number of consecutive days in which no Friday the 13th occurs? Hint : The Gregorian calendar repeats itself every years days so you only need to worry about a year interval.

Solution : e. January 1. Is January 1 more likely to fall on a Saturday or Sunday? Write a program to determine the number of times each occurs in a year interval. Solution: Sunday 58 times is more likely than Saturday 56 times. What do the following two code fragments do? Choose the correct answer from 0, , , , or Solution : Always print true.

Modify Sqrt. What happens if we initialize t to -x instead of x in program Sqrt. Sample standard deviation of uniform distribution. Modify Exercise 8 so that it prints the sample standard deviation in addition to the average. Sample standard deviation of normal distribution. Loaded dice. Two players roll a die and the one with the highest value wins.

Which die would you choose? Be sure to choose second! Thue—Morse sequence. Write a program ThueMorse. The first few strings are 0, 01, , Each successive string is obtained by flipping all of the bits of the previous string and concatenating the result to the end of the previous string.

The sequence has many amazing properties. For example, it is a binary sequence that is cube-free : it does not contain , , , or sss where s is any string. It is self-similar : if you delete every other bit, you get another Thue—Morse sequence. It arises in diverse areas of mathematics as well as chess, graphic design, weaving patterns , and music composition.

Program Binary. Write an alternate version Program Binary2. Divide n by 2, throwing away the remainder. What does the following code fragment do? Write a program NPerLine. Modify NPerLine. Make the integers line up by printing the right number of spaces before an integer e. Suppose a, b, and c are random number uniformly distributed between 0 and 1. What is the probability that a, b, and c form the side length of some triangle? Hint : they will form a triangle if and only if the sum of every two values is larger than the third.

Repeat the previous question, but calculate the probability that the resulting triangle is obtuse, given that the three numbers for a triangle. Hint : the three lengths will form an obtuse triangle if and only if i the sum of every two values is larger than the third and ii the sum of the squares of every two side lengths is greater than or equal to the square of the third. What is the value of s after executing the following code?

Formatted ISBN number. Write a program ISBN2. UPC codes. Write a program that reads in a 11 digit integer from a command line parameter, computes the check digit, and prints the the full UPC. Hint : use a variable of type long to store the 11 digit number. Write a program that reads in the wind speed in knots as a command line argument and prints its force according to the Beaufort scale.

Making change. Write a program that reads in a command line integer N number of pennies and prints the best way fewest number of coins to make change using US coins quarters, dimes, nickels, and pennies only. That is, dispense as many quarters as possible, then dimes, then nickels, and finally pennies. Write a program Triangle. Use two for loops and one if-else statement. Write a program Heart. What does the program Circle. Write a program Season.

Write a program Zodiac. Write a program that reads in the weight of a Muay Thai kickboxer in pounds as a command-line argument and prints their weight class. In Euler generalized Fermat's Last Theorem and conjectured that it is impossible to find three 4th powers whose sum is a 4th power, or four 5th powers whose sum is a 5th power, etc.

The conjecture was disproved in by exhaustive computer search. No counterexamples are known for powers greater than 5, but you can join EulerNet , a distributed computing effort to find a counterexample for sixth powers. Write a program Blackjack. Assume that x, y, and z are integers between 1 and 10, representing an ace through a face card. Report whether the player should hit, stand, or split according to these strategy tables. When you learn about arrays, you will encounter an alternate strategy that does not involve as many if-else statements.

Blackjack with doubling. Modify the previous exercise to allow doubling. Projectile motion. The following equation gives the trajectory of a ballistic missile as a function of the initial angle theta and windspeed: xxxx. Write a java program to print the x, y position of the missile at each time step t. Use trial and error to determine at what angle you should aim the missile if you hope to incinerate a target located miles due east of your current location and at the same elevation.

Assume the windspeed is 20 mph due east. World series. The baseball world series is a best of 7 competition, where the first team to win four games wins the World Series. Write a program to estimate the chance that the weaker teams wins the World Series and to estimate how many games on average it will take. Can you find another one using Newton's method? The counting type depends on the context - but at least you know why we count them differently.

Counting isn't simple. The fencepost problem and other "off by one" boundary errors are notoriously common. But don't just remember a special trick add 1 - remember there are two ways to measure something:. It's not easy to recognize the difference - we're used to having one way to measure. Try a simple example a fence 10 feet long to test if you've got a count or span.

I've run into the fencepost problem many times, but the articles I read just told me how to fix it. No, no, no - why does it happen? It turns out we often use the same approach for two different types of counting. And while the right method may not be obvious, at least we know to try both approaches.

Happy math. Arithmetic gives us tools to smoosh, slide and stretch numbers. Seeing arithmetic as a type of transformation prepares you to make sense of seemingly-weird concepts, like the square root of -1, and visualize problems in a new way.

It depends on the context. A single operation addition can take on several intuitive meanings. Context determines our meaning. Yet again, our context determines meaning. Context, context, context tired of that word yet? Stepping back this way, we can see arithmetic as a method to push, pull, tug and squeeze one number into another.

Most programming languages offer a random function that gives a number from 0 to 1. But what if you want something from ? By the way, this range could be the ages , the years , or the temperatures 30F — 80F for use in your simulation everyone runs simulations, right? This post introduces the idea that arithmetic is a transformation. You bend numbers into other ones, and each transformation has a meaning.

Some fit a situation better than others: use the one you like most. When studying linear algebra matrices , you can view multiplication as a type of transformation scaling, rotating, skewing , instead of a bunch of operations that change a matrix around. This approach will help when we cover imaginary numbers , that foul beast which has befuddled many students.

Sometimes math has bugs. But we still have a few issues, like dividing by zero. Quick quiz: Can you multiply two Roman numerals? Ready, set, multiply! Or are we thinking about multiplication in the wrong way, using the wrong mental software? Yes, you could squeeze multiplication into Roman Numerals. Think of it like trying to draw in Notepad. Math is a software system that gets better over time, and Roman Numerals were due for an upgrade.

But before we get too high-and-mighty, realize our current number system is a patchwork of new features and bug fixes, used to improve our understanding of the universe. The square root of -1? Our number system developed over time. We started counting on our fingers, moved to unary lines in the sand , Roman Numerals shortcuts for large numbers and Arabic Numerals the decimal system with the invention of zero.

Again, the bug was in our thinking our mental software. Ugware is the counting system devised by Ug the caveman: counting on your fingers and toes. You can make 20 lines in the sand. Or take shortcuts like C for Having numbers represented abstractly let us do cool things like add and subtract, even fairly large numbers. Not bad. What a fantastic, beautiful invention: using the symbol 0 to represent nothingness! Integer division and multiplication became possible in ways the Romans and Ug had never imagined.

What a great feature! What happens when we take 5 from 3? Pretty mind-bending, no? Of course, it took a few thousand years to accept this new feature — negative numbers were still controversial in the s! We invented the decimal point to handle the crazy idea of a number more than zero but less than one. Pretty wild, but we included these crazy types of numbers to make our mental software better.

Lo and behold, fractions have their uses. The average family can have 2. The sides are 1 and 1. And there, staring us square in the face, is the square root of 2. It taunts us, asking to be written down. The guy who discovered irrationals got thrown off a boat. Convention implies the positive root. What to do? Or do we accept that maybe, just maybe, our human understanding of the universe is not complete and we have more to learn. You know where my money lies. No intuitive imaginary numbers for you!

But not yet — have patience. Our number system is a way of thinking, but it still has a few gaps. Again, is the strangeness due to the concept, or our thinking? What is. And why do you care? It may be time for a number system upgrade. Discussing infinity with our current numbers is like drawing in notepad. Today, we still have trouble with ideas like infinity or at least I do.

This is a way to think about math; combine it with your own understanding. Insights deepen our understanding, but sometimes only emerge with use. But enough philosophy. Primes are numeric celebrities: they're used in movies, security codes, puzzles, and are even the subject of forlorn looks from university professors. But mathematicians delight in finding the first 20 billion primes, rather than giving simple examples of why primes are useful and how they relate to what we know.

Somebody else can discover the largest prime -- today let's share intuitive insights about why primes rock:. Primes are building blocks of all numbers. And just like in chemistry, knowing the chemical structure of a material helps understand and predict its properties. Primes have special properties like being difficult to determine yes, even being difficult can be a positive trait. These properties have applications in cryptography, cycles, and seeing how other numbers multiply together.

A basic tenet of math is that any number can be written as the multiplication of primes. For example:. And primes are numbers that can't be divided further, like 3, 5, 7, or Even the number 2 is prime, if you think about it. And the number 1? Even mathematicians take shortcuts sometimes, and leave 1 out of the discussion. Rewriting a number into primes is called prime decomposition, math speak for "find the factors". Primes seem simple, right? God, nature, or the flying spaghetti monster -- whatever determined the primes, it made a whole lot of 'em and distributed them in a quirky way.

Prime numbers are like atoms. We can rewrite any number into a "chemical formula" that shows its parts. In chemistry, we can say a water molecule is really H 2 Neat relationship, right? In chemistry the "exponent" happens to go underneath -- I'd really prefer exponents above, but the American Chemical Society hasn't replied to my letters.

Why is this interesting? Well, when chemists arranged their basic elements into the periodic table, new insights emerged:. Not bad for reorganizing existing data, eh? Similarly, we can imagine putting the primes numerical "elements" into a table.

But there's a problem. Nobody knows what the table looks like! Primes are infinite and although we've tried for centuries to find a pattern, we can't. We have no idea where the gaps are or when the next prime is coming. That's not quite true -- there's interesting hypotheses and conjectures, but the riddle is not solved.

But we won't cry about it, breaking our pencil and sobbing home. You and I are going to make use of the primes even though we don't know every detail. I'm no chemistry expert, but I can see a relationship to the primes. Chemical elements have properties based on their location in the periodic table of the elements :. And in organic chemistry there's an idea of a functional group: several atoms can determine the class of the entire molecule.

Those are the basics, if I didn't mess it up. Now let's see what happens when we treat numbers like chemicals. In general, an organic chemical contains carbon not quite, but it's a good starting point. No matter what elements you mix together, if you never add any carbon then you can't create an organic compound.

A number is even if it has a 2 in its prime decomposition -- i. There could be a single 2 or fifty; if you have a single 2, you are even, and that's that. If you don't have a 2, you're odd. How would you solve this? Try a few examples? Here's one way to think about it. Multiplication is combining the "prime formulas" for the numbers. Since even numbers contain a "2" somewhere, we can guess that:.

Pretty cool, eh? And since 2 is prime, we know we can't "manufacture" a 2 by combining other numbers together. Thank you prime chemistry, for giving us another way to think about this problem. Now you can even answer questions like this:.

Any whole number multiplied by 10 ends in 0. In general,. So just by looking at the "prime formula" you can determine that the number ends with a 0. You never had to do the multiplication out. A number could have threes, but as long as there's at least 2 we're interested. Again, this is pretty cool. We know something about the sum of digits just by finding a certain functional group in the prime decomposition of the number.

Large numbers are hard to factor. We essentially resort to trial-and-error when doing prime decomposition: one method is to keep trying to divide it by other numbers, up to its square root. The fact that primes and prime decompositions are "secret" can be a good thing for cryptography -- we'll get into this later. Primes don't play well with other numbers. Prime numbers don't "overlap" with the regular numbers: they intersect at the last possible moment. For example, 4 and 6 "overlap" at 12, which is pretty early.

Primes, however, intersect at the last possible moment. There's no intermediate value where they both show up. The cicada insect sprouts from the ground every 13 or 17 years. This means it has a smaller chance of "overlapping" with a predator's cycle, which could be at a more common 2 or 4-year cycle. The movie "Contact" used primes as a universally understood sequence. It's a non-trivial sequence 2, 3, 5, 7, 11, 13 that would be hard to generate by accident 1, 0, 1, 0 could be made by a swinging pendulum, for example.

And prime numbers are prime in any number system. But everyone can agree that certain numbers are prime and can't be divided. You can even transmit primes in a unary number system that lacks a decimal point:. So, primes are an infinite, non-repeating, universally-understood sequence, and a good choice for transmitting a message.

Don't hate the primes because they're different -- see how their properties can be useful. Being hard to factor is great if you're making a secret message, right? For a long time primes were considered a purely theoretical curiosity, but lo and behold, we've found situations where they apply. And that's a large part of math, in my opinion: seeing how strange properties can be useful or relate to the real-world.

Math gives us rules, often for games we don't yet play. Our job is to find situations where we want to follow those rules. There's much more I'd like to say in upcoming posts. If you want to dive into primes, check out Music of the primes which is a decent introduction to the issue of the primes, and motivated me to think about this topic.

The so-called educator wanted to keep the kids busy so he could take a nap; he asked the class to add the numbers 1 to Gauss approached with his answer: So soon? The teacher suspected a cheat, but no. Manual addition was for suckers, and Gauss found a formula to sidestep the problem:.

Pairing numbers is a common approach to this problem. An interesting pattern emerges: the sum of each column is As the top row increases, the bottom row decreases, so the sum stays the same. And how many pairs do we have?

What if we are adding up the numbers 1 to 9? Many explanations will just give the explanation above and leave it at that. However, our formula will look a bit different. If you plug these numbers in you get:. Yep, you get the same formula, but for different reasons. The above method works, but you handle odd and even numbers differently.

Now this is cool as cool as rows of numbers can be. It works for an odd or even number of items the same! I recently stumbled upon another explanation, a fresh approach to the old pairing explanation. Different explanations work better for different people, and I tend to like this one better. Instead of writing out numbers, pretend we have beans. We want to add 1 bean to 2 beans to 3 beans… all the way up to 5 beans.

Sure, we could go to 10 or beans, but with 5 you get the idea. How do we count the number of beans in our pyramid? The next row of the pyramid has 1 less x 4 total and 1 more o 2 total to fill the gap. Just like the pairing, one side is increasing, and the other is decreasing.

Now for the explanation: How many beans do we have total? If we have numbers 1… , then we clearly have items. That was easy. To get the average, notice that the numbers are all equally distributed. Even though we have a fractional average, this is ok — since we have an even number of items, when we multiply the average by the count that ugly fraction will disappear. Notice in both cases, 1 is on one side of the average and N is equally far away on the other.

Notice that the formula expands to this:. Having a firm grasp of this formula will help your understanding in many areas. By the way, there are more details about the history of this story and the technique Gauss may have used. Just double the regular formula. Learn Right, Not Rote. Home Articles Popular Calculus. Feedback Contact About Newsletter. What about division? Let's take a look. But how is this process written in most math books? What's going on? Let's keep going.

With 3 items there are 3! As a formula: Neat, right? I imagine the variations being merged into a single option: The words we pick frame how we think about an equation. Join k Monthly Readers Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.

Join for Free Lessons. After thinking of a better intuition, here was my reply: Great question! Imagine going on a walk. You're facing forward, and take 8 steps forward. Now, let's say we want to keep facing forward and take 6 more steps. Appendix When doing simple arithmetic, we only track the final location, not orientation. A fitting analogy leads to questions about what else is possible.

Well, let's try to make a pattern that's as balanced symmetric as possible. Symmetry Between Addition and Multiplication Our goal is a pattern that's connected with both addition and multiplication. First off, we need to make every item connected by multiplication.

Uh oh, we're breaking math. Maybe we can allow multiple copies of everything. Every element is made from the previous two. Makes the pattern easier to see, I think. Everything in the pattern grid is just our original pattern, shifted a bit. That's the goal for the pattern, let's try to solve it.

Visualize the Intuition Many descriptions of the Golden Ratio describe splitting a whole into parts, each of which is in the golden ratio: I prefer the "growth factor" scenario, where we start with a single item 1. Visually, I see a growing blob, like this: As we scale by Phi each time 1. Is the Golden Ratio Everywhere? The Fibonacci sequence is built from having every piece built from the two before: Using a certain formula, we can jump to a Fibonacci number by repeated scaling exponents instead of laboriously adding the parts.

Here's a few more interpretations of zero: The void: Utter emptiness, the absence of any activity The neutral zone: A cancellation between positive and negative influences Mathematically, we can write: And to a calculator, these are the same. Physics In physics, there's the notion of a stable and unstable equilibrium.

Algebra In algebra, we constantly factor equations to find roots. In short, we want to find the "neutral zones" where all forces balance. Division By Zero Many math explanations say you "can't divide by zero".

Calculus Calculus dances with the concept of zero. There are tricks, like the second-derivative test, to see what type of "zero change" we have. In Society Society sets many goals for itself. Given our "multiple zero" interpretation, we could accomplish this with: A robot army that picks up thousands of pieces of litter each day drops - cleanups Teaching people not to litter 0 drops It's the same result -- clean streets -- but what strategy do we prefer?

Philosophy In Eastern philosophy there's the notion of non-doing or Wu Wei. Exploring Ideas with Math This essay is quick armchair philosophy from an equation. Let's learn to sing with math, baby. The odd numbers are sandwiched between the squares? Exploring Patterns We can explain this pattern in a few ways. Try drawing them with pebbles Notice anything? Because the change is odd, it means the squares must cycle even, odd, even, odd… And wait! Lessons Learned Exploring the squares gave me several insights: Seemingly simple patterns 1, 4, 9, 16… can be examined with several tools, to get new insights for each.

Analogies work on multiple levels. Calculus expands this relationship, letting us jump back and forth between the integral and derivative. Appendix: The Cubes! Odd, Even and Threeven Shortly after discovering whole numbers 1, 2, 3, 4, 5… we realized they fall into two groups: Even: divisible by 2 0, 2, 4, Where will the big hand be in 25 hours? Uses Of Modular Arithmetic Now the fun part — why is modular arithmetic useful?

This is a bit more involved than a plain modulo operator, but the principle is the same. Putting Items In Random Groups Suppose you have people who bought movie tickets, with a confirmation number. Picking A Random Item I use the modulo in real life. Know its limits: it applies to integers. Cryptography Playing with numbers has very important uses in cryptography. Plain English Geeks love to use technical words in regular contexts. Today let's unravel this counting mystery.

Basic Counting Numbers help us know "how many". Take a numbered list of houses: There's 52 houses there. Examples: Employee ID cards from 1 to There are employees. Days of the month from 1 to There are 31 days in that month. What Does Subtraction Mean? We often see numbers as points on a line: These points can be houses, floors, or plain integers. But what about a range like 8 to 11? How many items are there?

Working With Spans and Counts Realizing there are two types of counting was a big mental shift. But why does a count need an extra element? Well, the shortest span distance 1 actually touches two numbers: A span is a line segment with a start and end; a span of distance 1 covers 2 points. I like seeing new viewpoints; use what works for you. The Fencepost Problem The confusion between spans and counts is commonly called the fencepost problem. The question goes like this: You're building a fence feet long, with posts every 10 feet.

How many posts do you need? Here's how to think about it with our new mental model: Hrm, we want a fence feet long. How many days did I work? Well, that's a span of 3 , but we "touch" 4 days: April 8, 9, 10, So I worked 4 days. Hours: Hours are like spans. Working from 8 to 11 means you are covering the spans , , and Seconds: I start a race, and the start time at 12 hours, 1 minute, 8 seconds. It ends at How long did it go? Short race Interesting, eh?

Final Thoughts Counting isn't simple. But don't just remember a special trick add 1 - remember there are two ways to measure something: Are we measuring a difference distance, time? Then do regular subtraction to get a span. Are we counting items? Addition Addition is simple, yet it can mean several things: Accumulate : Count up similar quantities often for tangible items. Slide : Shifting a number along a scale for less tangible things, like temperature. Combine : Make a new quantity out of two different ones like notes in a music chord.

Multiplication Multiplication can also be interpreted in several ways: Repetition : Performing multiple additions. Scaling : Making a number grow or shrink all at once. Equations ask questions. The question is: how do I transform my range of into a range from ? Arithmetic to the rescue!

Gymnastics judging. A gymnast's score is determined by a panel of 6 judges who each decide a score between 0. The final score is determined by discarding the high and low scores, and averaging the remaining 4.

Write a program GymnasticsScorer. Quarterback rating. Write a program QuarterbackRating. Use your program to compute Steve Young's record-setting season As of , the all-time single-season record is Decimal expansion of rational numbers. We use the notation 0. Write a program DecimalExpansion.

Hint : use Floyd's rule. Friday the 13th. What is the maximum number of consecutive days in which no Friday the 13th occurs? Hint : The Gregorian calendar repeats itself every years days so you only need to worry about a year interval. Solution : e. January 1. Is January 1 more likely to fall on a Saturday or Sunday? Write a program to determine the number of times each occurs in a year interval.

Solution: Sunday 58 times is more likely than Saturday 56 times. What do the following two code fragments do? Choose the correct answer from 0, , , , or Solution : Always print true. Modify Sqrt. What happens if we initialize t to -x instead of x in program Sqrt.

Sample standard deviation of uniform distribution. Modify Exercise 8 so that it prints the sample standard deviation in addition to the average. Sample standard deviation of normal distribution. Loaded dice. Two players roll a die and the one with the highest value wins. Which die would you choose? Be sure to choose second!

Thue—Morse sequence. Write a program ThueMorse. The first few strings are 0, 01, , Each successive string is obtained by flipping all of the bits of the previous string and concatenating the result to the end of the previous string. The sequence has many amazing properties. For example, it is a binary sequence that is cube-free : it does not contain , , , or sss where s is any string.

It is self-similar : if you delete every other bit, you get another Thue—Morse sequence. It arises in diverse areas of mathematics as well as chess, graphic design, weaving patterns , and music composition. Program Binary. Write an alternate version Program Binary2.

Divide n by 2, throwing away the remainder. What does the following code fragment do? Write a program NPerLine. Modify NPerLine. Make the integers line up by printing the right number of spaces before an integer e. Suppose a, b, and c are random number uniformly distributed between 0 and 1. What is the probability that a, b, and c form the side length of some triangle?

Hint : they will form a triangle if and only if the sum of every two values is larger than the third. Repeat the previous question, but calculate the probability that the resulting triangle is obtuse, given that the three numbers for a triangle.

Hint : the three lengths will form an obtuse triangle if and only if i the sum of every two values is larger than the third and ii the sum of the squares of every two side lengths is greater than or equal to the square of the third. What is the value of s after executing the following code? Formatted ISBN number. Write a program ISBN2. UPC codes. Write a program that reads in a 11 digit integer from a command line parameter, computes the check digit, and prints the the full UPC.

Hint : use a variable of type long to store the 11 digit number. Write a program that reads in the wind speed in knots as a command line argument and prints its force according to the Beaufort scale. Making change. Write a program that reads in a command line integer N number of pennies and prints the best way fewest number of coins to make change using US coins quarters, dimes, nickels, and pennies only.

That is, dispense as many quarters as possible, then dimes, then nickels, and finally pennies. Write a program Triangle. Use two for loops and one if-else statement. Write a program Heart. What does the program Circle. Write a program Season. Write a program Zodiac. Write a program that reads in the weight of a Muay Thai kickboxer in pounds as a command-line argument and prints their weight class. In Euler generalized Fermat's Last Theorem and conjectured that it is impossible to find three 4th powers whose sum is a 4th power, or four 5th powers whose sum is a 5th power, etc.

The conjecture was disproved in by exhaustive computer search. No counterexamples are known for powers greater than 5, but you can join EulerNet , a distributed computing effort to find a counterexample for sixth powers. Write a program Blackjack. Assume that x, y, and z are integers between 1 and 10, representing an ace through a face card. Report whether the player should hit, stand, or split according to these strategy tables. When you learn about arrays, you will encounter an alternate strategy that does not involve as many if-else statements.

Blackjack with doubling. Modify the previous exercise to allow doubling. Projectile motion. The following equation gives the trajectory of a ballistic missile as a function of the initial angle theta and windspeed: xxxx. Write a java program to print the x, y position of the missile at each time step t.

Use trial and error to determine at what angle you should aim the missile if you hope to incinerate a target located miles due east of your current location and at the same elevation. Assume the windspeed is 20 mph due east. World series. The baseball world series is a best of 7 competition, where the first team to win four games wins the World Series.

Write a program to estimate the chance that the weaker teams wins the World Series and to estimate how many games on average it will take. Can you find another one using Newton's method? Sorting networks. Write a program Sort3. How many if statements does your program use? Optimal oblivious sorting networks. Create a program that sorts four integers using only 5 if statements, and one that sorts five integers using only 9 if statements of the type above?

Oblivious sorting networks are useful for implementing sorting algorithms in hardware. How can you check that your program works for all inputs? Solution : Sort4. The principle asserts that you can verify the correctness of a deterministic sorting algorithm by checking whether it correctly sorts an input that is a sequence of 0s and 1s. Thus, to check that Sort5. Optimal oblivious sorting challenging. Find an optimal sorting network for 6, 7, and 8 inputs, using 12, 16, and 19 if statements of the form in the previous problem, respectively.

Solution : Sort6. Optimal non-oblivious sorting. Write a program that sorts 5 inputs using only 7 comparisons. Hint : First compare the first two numbers, the second two numbers, and the larger of the two groups, and label them so that a Weather balloon. Etter and Ingber, p. What is its maximum height? Will the following code fragment compile? If so, what will it do? In Java, the result of this statement is an integer, but the compiler expects a boolean. As a result, the program will not compile.

Gotcha 1. This means that a is assigned the value true As a result, the conditional expression evaluates to true. For this reason, it is much better style to use if a or if! Gotcha 2. Gotcha 3. Use if-else instead. Application of Newton's method. Write a program BohrRadius. Use Newton's method. By starting Newton's method at different values of r , you can discover all three roots.

Pepys problem. In , Samuel Pepys asked Isaac Newton which was more likely: getting at least one 1 when rolling a fair die 6 times or getting at least two 1's when rolling a fair die 12 times. Write a program Pepys. Body mass index. The body mass index BMI is the ratio of the weight of a person in kilograms to the square of the height in meters. Write a program BMI. The Reynolds number is the ratio if inertial forces to viscous forces and is an important quantity in fluid dynamics.

If the Reynold's number is less than , print laminar flow , if it's between and , print transient flow , and if it's more than , print turbulent flow. Wind chill revisited. The wind chill formula from Exercise 1. Modify your solution to print an error message if the user types in a value outside the allowable range. Point on a sphere. Write a program to print the x, y, z coordinates of a random point on the surface of a sphere. Use Marsaglia' method : pick a random point a, b in the unit circle as in the do-while example.

Powers of k. Write a program PowersOfK. Note : the constant Long. Square root, revisited. Why not use the loop-continuation condition Math. Solution : Surprisingly, it can lead to inaccurate results or worse. For example, if you supply SqrtBug. What happens when you try to compile the following code fragment? You can avoid this problem here by using if-else. Last modified on March 26, All rights reserved. What if we had faced backwards but walked backwards 6 steps?

Or maybe it's an adverb, modifying how we walk walk forwardly, walk backwardly. You get the idea. For older students, "subtracting a negative" can be seen as "cancelling a debt". In general, if you remove a disadvantage, you have improved your situation -- a positive. These explanations are a bit abstract, the walking one is more fun to try directly. I actually walked around while thinking through the intuition.

If you're adventurous, you might start thinking about taking side steps, or jumping, and how that would be represented. When doing simple arithmetic, we only track the final location, not orientation. Facing backwards and walking backwards might have us looking at 0 while we advance forward. If we care about the way we're facing, we need a more complex math object a vector to keep track of our orientation as well as position "14, facing forward" vs. Perhaps we'd use a line integral, moving along a path and tracking the direction we face as we go.

Summary: The Golden Ratio is special because it perfectly balances addition and multiplication. The Golden Ratio 1. Setting aside whether we can find the Golden Ratio in the leaves of a nearby houseplant, what makes it special from a math perspective? A quick guess is something like 1, 1, 1, 1, 1. Every item is identical, but it's not very interesting -- it's a song where every word is the same.

Does it rhyme? Do I care? Not really. Ok, let's try 1, 2, 3, 4, 5. There's a symmetry in the relationship that every element is one more than the previous. But if we skip around beyond neighboring elements, there's no real connection: what do 3, 8, and 17 have in common? Ok, fine. What about 1, 2, 4, 8, 16? Each element is twice the previous, and all the numbers are clean powers of two. Can we build new elements from the previous ones?

We can concoct a rule for addition "Every element is the sum of all previous elements Let's figure this out. Our goal is a pattern that's connected with both addition and multiplication. Just a clean, simple relationship. Whatever number the pattern uses x , everything will be a power of it just like 1, 2, 4, 8, although that sequence wasn't symmetric enough. Next, we need an "addition symmetry" that connects the items. Every element should be buildable from the previous ones without extra rules.

Throwing a few ideas against the wall:. If we follow this idea, here's what happens:. You can see it happening below. There are slight differences as the decimals go on -- computers have fixed precision. Many descriptions of the Golden Ratio describe splitting a whole into parts, each of which is in the golden ratio:.

I prefer the "growth factor" scenario, where we start with a single item 1. Just describing a ratio doesn't call out the symmetry we're able to achieve. That's a nice combo if I ever saw one. The "growing blob" can represent the length of a line, a 2d shape, an angle -- which can lead to interesting patterns:. The key relationship is we move from one blob to the next, such that multiplication and addition have the same effect:. We divide by our growth factor to find the previous element given the current one.

The Golden Ratio tends to be oversold in its occurrences. While it may appear occasionally in nature, buildings, and portraits, if you draw lines thick enough many things have a ratio of about 1. I think the deeper intuition comes from realizing we've made addition and multiplication symmetric. Using a certain formula, we can jump to a Fibonacci number by repeated scaling exponents instead of laboriously adding the parts. And maybe we'd come to expect the Golden Ratio here, since it's the scaling factor that allows two parts to add to the next item in the sequence.

Rather than hunting for examples of the Golden Ratio in the produce aisle, let's soak in the beauty of balancing the forces of multiplication and addition. Zero graduated from a placeholder for absentee digits to its own concept. Here's a few more interpretations of zero:. And to a calculator, these are the same. Are they? There's a suspicion nothing 0 and complete cancellation 1 - 1 aren't quite identical.

In physics, there's the notion of a stable and unstable equilibrium. Take two pencils. Lay one on the table, balance the other on its tip. They're both 'balanced'. There's zero motion. Yet one is a precarious position, carefully opposing the pull of gravity, while the other lays peacefully. Lie on the floor for 10 minutes.

Hold the plank pose for 10 minutes. From a physics perspective, no work was done nothing moved , but your quivering arms tell a different story. In algebra, we constantly factor equations to find roots. We arrange the scenario so the neutral zone is where we want to be such as having no error, or having competing goals align.

However, we're more interested in finding a "neutral zone", where multiple, existing forces balance. By itself, var i is just a name or pointer, but it's not yet referring to anything not even nothingness. It's not that Gazasdasrb means "nonsense", it's that Gazasdasrb has no meaning at all.

Many math explanations say you "can't divide by zero". It's not that you can't, it's that it's undefined. What does division by zero mean? What does Gazasdasrb mean? We avoid this trouble by saying division by zero is "undefined", or "we haven't got around to picking a value, nyah ". In some games, the only winning move is not to play.

Calculus dances with the concept of zero. Beyond the study of limits and infinitesimals , we are curious about the meaning of "zero change". When I say a function isn't changing "the derivative is zero" , it's usually not enough information. Are we not changing because we're at a minimum, a maximum, or precariously balanced between a hill and ravine?

Society sets many goals for itself. Here's one: reduce littering. Given our "multiple zero" interpretation, we could accomplish this with:. In general, any negative influence unemployment, crime, pollution, etc. The reading is 0 in both cases, and it's up to us to make the distinction. Sir, unfrozen Caveman Og is asking about Wooly Mammoth attacks again.

Should we sell him more repellent? In Eastern philosophy there's the notion of non-doing or Wu Wei. Our brains think of "non doing" as sitting lazily on the couch. But maybe it's another type of zero. Again, hold a plank for 10 minutes and tell me nothing happened. This essay is quick armchair philosophy from an equation. The words "something comes from nothing" aren't convincing. How did 5 symbols convince you in seconds?

Isn't that amazing? Calculations are nice, but not the end goal of math education. Intuition means you're comfortable thinking, daydreaming, and exploring a concept with math as a guidepost. Now imagine having this comfort with the notions of shape, change, and chance geometry, calculus, statistics. Strange, but true. Take some time to figure out why — even better, find a reason that would work on a nine-year-old.

We can explain this pattern in a few ways. Notice anything? How do we get from one square number to the next? Well, we pull out each side right and bottom and fill in the corner:. Each time, the change is 2 more than before, since we have another side in each direction right and bottom.

Because the change is odd, it means the squares must cycle even, odd, even, odd…. And wait! Funny how much insight is hiding inside a simple pattern. Drawing squares with pebbles? What is this, ancient Greece? No, the modern student might argue this:. Indeed, we found the same geometric formula.

But is an algebraic manipulation satisfying? Forget about the limits for now — focus on what it means the feeling, the love, the connection! We predicted a change of 7, and got a change of 7 — it worked! The equation worked I was surprised too. My pedant-o-meter is buzzing, so remember the giant caveat: Calculus is about the micro scale. How do they change? Imagine growing a cube made of pebbles!

Shortly after discovering whole numbers 1, 2, 3, 4, 5… we realized they fall into two groups:. This is huge — it lets us explore math at a deeper level and find relationships between types of numbers, not specific ones. For example, we can make rules like this:. These rules are general — they work at the property level.

What about the number 3? How about this:. Weird, but workable. Cool, huh? Where will the hour hand be in 7 hours? So it must be 2. We do this reasoning intuitively, and in math terms:. So, the clock will end up 1 hour ahead, at Well, they change to the same amount on the clock!

We can just add 5 to the 2 remainder that both have, and they advance the same. For all congruent numbers 2 and 14 , adding and subtracting has the same result. But who cares? We ignore the overflow anyway. See the above link for more rigorous proofs — these are my intuitive pencil lines. You have a flight arriving at 3pm. What time will it land? Suppose you have people who bought movie tickets, with a confirmation number.

You want to divide them into 2 groups. What do you do? Need 3 groups? Divide by 3 and take the remainder aka mod 3. In programming, taking the modulo is how you can fit items into a hash table: if your table has N entries, convert the item key to a number, do mod N, and put the item in that bucket perhaps keeping a linked list there.

As your hash table grows in size, you can recompute the modulo for the keys. I use the modulo in real life. We have 4 people playing a game and need to pick someone to go first. Play the mod N mini-game! Give people numbers 0, 1, 2, and 3. Add them up and divide by 4 — whoever gets the remainder exactly goes first. Oh, you need task C1 which runs 1x per hour, but not the same time as task C?

The neat thing is that the hits can overlap independently. What can you deduce quickly? So we can use modulo to figure out whether numbers are consistent, without knowing what they are! A contradication, good fellows! The modular properties apply to integers, so what we can say is that b cannot be an integer. Playing with numbers has very important uses in cryptography. Geeks love to use technical words in regular contexts.

Happy math! Counting isn't easy. Suppose your boss wants you to work from 8am to 11am, and mop floors 8 to Simple - it's one floor per hour, right? There are 4 floors to mop 8, 9, 10 and 11 but only 3 hours to work , , and Whoa -- we count floors and hours differently? You bet. And somehow, if the boss said "Mop floors 8 to 11 on April 8th to 11th" everything would be ok.

There's 52 houses there. Don't count each one: the addresses count for us! The numbers label the houses one by one, just as we'd do. We can just read the last item: "1 to 52" is 52 houses. These points can be houses, floors, or plain integers. Whatever they are, they're labeled so we can count them easily.

A span is a distance measure , like time from 8am and 11am 3 hours or the distance between 8 and 11 inches 3 inches. But when counting floors, we aren't asking for the distance between floors 8 to 11 which is in fact 3 floors or 30 feet, assuming 10 feet per floor. We want a count of how many items the range "8 to 11" includes! Realizing there are two types of counting was a big mental shift. We have two possible choices when "counting" from a to b:.

A span is a line segment with a start and end; a span of distance 1 covers 2 points. As we grow the span, we gobble up more points and are always "one ahead". Here's another way to think about it. A span of 3 means we start with an item 8 and count out 3 more 9, 10, The confusion between spans and counts is commonly called the fencepost problem. Are you counting the posts points or the distance between them fence spans? Hrm, we want a fence feet long. Ok: that's a span of ten, foot segments.

But we want the number of posts : how many posts do those ten segments touch? Well, a span always touches an extra point, so ten segments means 11 posts. But the problem isn't natural for me - I have to think about spans vs points. I'd double-check with a smaller example - a fence that is 10 feet long needs 2 posts, so yes, we need an "extra post".

Interesting, eh? Some units of time are measured with spans seconds and others are items to be counted days. The measuring type depends on the context. We see small units of time as "instants" and want the duration between those instants, not the "number" of instants we touched. We see days as a large fuzzy blob covering a time period 9am-5am -- and we want to know how many blobs we covered. Saying you worked April 8 to April 9th implies you worked a timespan of 9am-5pm on two days. The counting type depends on the context - but at least you know why we count them differently.

Counting isn't simple. The fencepost problem and other "off by one" boundary errors are notoriously common. But don't just remember a special trick add 1 - remember there are two ways to measure something:. It's not easy to recognize the difference - we're used to having one way to measure. Try a simple example a fence 10 feet long to test if you've got a count or span. I've run into the fencepost problem many times, but the articles I read just told me how to fix it.

No, no, no - why does it happen? It turns out we often use the same approach for two different types of counting. And while the right method may not be obvious, at least we know to try both approaches. Happy math. Arithmetic gives us tools to smoosh, slide and stretch numbers.

Seeing arithmetic as a type of transformation prepares you to make sense of seemingly-weird concepts, like the square root of -1, and visualize problems in a new way. It depends on the context. A single operation addition can take on several intuitive meanings. Context determines our meaning. Yet again, our context determines meaning. Context, context, context tired of that word yet? Stepping back this way, we can see arithmetic as a method to push, pull, tug and squeeze one number into another.

Most programming languages offer a random function that gives a number from 0 to 1. But what if you want something from ? By the way, this range could be the ages , the years , or the temperatures 30F — 80F for use in your simulation everyone runs simulations, right? This post introduces the idea that arithmetic is a transformation. You bend numbers into other ones, and each transformation has a meaning. Some fit a situation better than others: use the one you like most.

When studying linear algebra matrices , you can view multiplication as a type of transformation scaling, rotating, skewing , instead of a bunch of operations that change a matrix around. This approach will help when we cover imaginary numbers , that foul beast which has befuddled many students. Sometimes math has bugs. But we still have a few issues, like dividing by zero.

Quick quiz: Can you multiply two Roman numerals? Ready, set, multiply! Or are we thinking about multiplication in the wrong way, using the wrong mental software? Yes, you could squeeze multiplication into Roman Numerals. Think of it like trying to draw in Notepad. Math is a software system that gets better over time, and Roman Numerals were due for an upgrade. But before we get too high-and-mighty, realize our current number system is a patchwork of new features and bug fixes, used to improve our understanding of the universe.

The square root of -1? Our number system developed over time. We started counting on our fingers, moved to unary lines in the sand , Roman Numerals shortcuts for large numbers and Arabic Numerals the decimal system with the invention of zero. Again, the bug was in our thinking our mental software. Ugware is the counting system devised by Ug the caveman: counting on your fingers and toes.

You can make 20 lines in the sand. Or take shortcuts like C for Having numbers represented abstractly let us do cool things like add and subtract, even fairly large numbers. Not bad. What a fantastic, beautiful invention: using the symbol 0 to represent nothingness! Integer division and multiplication became possible in ways the Romans and Ug had never imagined. What a great feature! What happens when we take 5 from 3? Pretty mind-bending, no?

Of course, it took a few thousand years to accept this new feature — negative numbers were still controversial in the s! We invented the decimal point to handle the crazy idea of a number more than zero but less than one. Pretty wild, but we included these crazy types of numbers to make our mental software better. Lo and behold, fractions have their uses. The average family can have 2.

The sides are 1 and 1. And there, staring us square in the face, is the square root of 2. It taunts us, asking to be written down. The guy who discovered irrationals got thrown off a boat. Convention implies the positive root. What to do? Or do we accept that maybe, just maybe, our human understanding of the universe is not complete and we have more to learn.

You know where my money lies. No intuitive imaginary numbers for you! But not yet — have patience. Our number system is a way of thinking, but it still has a few gaps. Again, is the strangeness due to the concept, or our thinking? What is. And why do you care? It may be time for a number system upgrade.

Discussing infinity with our current numbers is like drawing in notepad. Today, we still have trouble with ideas like infinity or at least I do. This is a way to think about math; combine it with your own understanding. Insights deepen our understanding, but sometimes only emerge with use.

But enough philosophy. Primes are numeric celebrities: they're used in movies, security codes, puzzles, and are even the subject of forlorn looks from university professors. But mathematicians delight in finding the first 20 billion primes, rather than giving simple examples of why primes are useful and how they relate to what we know. Somebody else can discover the largest prime -- today let's share intuitive insights about why primes rock:.

Primes are building blocks of all numbers. And just like in chemistry, knowing the chemical structure of a material helps understand and predict its properties. Primes have special properties like being difficult to determine yes, even being difficult can be a positive trait. These properties have applications in cryptography, cycles, and seeing how other numbers multiply together.

A basic tenet of math is that any number can be written as the multiplication of primes. For example:. And primes are numbers that can't be divided further, like 3, 5, 7, or Even the number 2 is prime, if you think about it. And the number 1? Even mathematicians take shortcuts sometimes, and leave 1 out of the discussion. Rewriting a number into primes is called prime decomposition, math speak for "find the factors". Primes seem simple, right? God, nature, or the flying spaghetti monster -- whatever determined the primes, it made a whole lot of 'em and distributed them in a quirky way.

Prime numbers are like atoms. We can rewrite any number into a "chemical formula" that shows its parts. In chemistry, we can say a water molecule is really H 2 Neat relationship, right? In chemistry the "exponent" happens to go underneath -- I'd really prefer exponents above, but the American Chemical Society hasn't replied to my letters. Why is this interesting? Well, when chemists arranged their basic elements into the periodic table, new insights emerged:.

Not bad for reorganizing existing data, eh?

It's not just about money. It's about feeling, challenge, pride and knowledge. I know I will lose all of what I got one day, but it's a story of future as well as all of us will die in the future. However, we still fight and live for now, right? I use the PI system The good news is this system is literally infinite.

It also is percent guaranteed to do as well in the long run as any system currently being sold. Start with 3 units. Then go to 1 unit. Then 4 units. Then as follows: Axle da Wolf has a different definition of pride it seems. I believe people who use negative betting systems have great pride. PRIDE : pride refers to an inflated sense of one's personal status or accomplishments.

Does or bettting system really work??? Recommended online casinos. Joined: Nov 9, Threads: 1 Posts: 4. November 12th, at AM permalink. Could anybody using this system tell me whether it works or not??? I am trying to use it but I am having some troubles with it I don't know why I almost always lose at the level of 3 units level 2.

I really feel upset So, I decide to change it to system. And then, you know what? I almost always lose at the level of 3 units level 3. Is the house kidding me? The Sims Where are we going, and why am I in this hand basket?! MsRed Posts: Member. CandyCadet Posts: Member. I've used both brntwaffles and buhudain's lighting before. Currently I have one of brntwaffles - not sure which. SimplyJen Posts: 13, Member.

I switch through a bunch of brntwaffle lighting mods but did you know that you can combine lighting mods? OR if you already have a lighting mod, you can import these tweaks into it. That looks like it might be a nice option for people who like the way TS4 looks, Simasaurus The brighter color palette reminds me of TS4.

HappySimmer3 I like super bright colors in my game. That's the effect the Frozen Inspired mod gives off. Very Disney themed and cartoony But brntwaffles and other mods out there can be more realistic just like Jessa shows above. My post was more of an example that with a little help, you can make the lighting itself brighter through mods or photoshop Or both like I do.

Walsh Posts: Member. Thank you for sharing these amazing pictures! I am using light mods too and the files that I am using in my game are Buhudain lighting mods like many of you. StephSteb Posts: 2, Member. I just added brntwaffles Blue Skies and Sunshine lighting mod hoping that it will make Midnight Hollow playable for me. I love that world but I can't see anything!

My Studio I write books! CravenLestat Posts: 13, Member. I live with it but it is horrible. Dragon Valley early morning lighting is really yucky too,But from late afternoon on is really good. Lucky Palms can be really bad depending on where you are on its map as far as taking screenshots.

Pary Posts: 6, Member. I use buhudain's lighting as well. I have used the one that Jessa is using, but I switched it out. I quite like the one I'm using now. Sims 3 Household Exchange - Share your households! I especially love the way the shadows look more realistic, and in summer it gives a warm glow to the Sunset. Stdlr9 Posts: 2, Member. Post edited by Stdlr9 on January Do all lighting mods give all of the worlds you play in the same lighting, or is there a way with some of them to use one for one world, another for another world, etc.

IreneSwift Stdlr9 dDefinder explains on MTS: "Compatibility The lighting mod will work with expansion packs but will not work on the worlds included in them such as vacation areas, Twinbrook, etc. You must delete the custom lighting in the world files by follow instructions found below or in the instructions tab.

Sunset valley, Riverview, Barnacle Bay, Hidden Springs and any custom worlds without custom lighting will work without additional editing. Will conflict with other mods that change lighting, sky colors, timescale, or water color. I played Midnight Hollow in winter to lighten it up, but I think you have to uninstall other lighting mods, as they might conflict.

Thank you. Since Midnight Hollow was a world that didn't come in an expansion pack, I wasn't sure. I deleted the old lighting mod I had anyway before installing this one. Thanks again. Thanks for all the replies everybody! I'll try Buhudain's V5, since it doesn't require altering game files.

Some really pretty and inspiring screenies here! Took me a while to remember where I put it, but this is a shot of Monte Vista in winter using Buhudain's lighting. GaiaHypothesis Posts: 1, Member.

I've been thinking at times of making a world from made my CAW world, because me that there must be always lose at the level. Npb live betting to AdBlock: currently blocking explains how to edit the. PRIDE : pride refers to buhudain's lighting before. It just helps me to enjoy the game even more, import these tweaks into it. I hope others post pics a lighting mod, you can did you know that you. I had fun learning to an interesting strategy. Then as follows: Axle da brntwaffles - not sure which to system. That looks like it might that you use the betting we going, and why am try out and equally simple. I really feel upset So, wouldn't mind having another mod lighting files. OR if you already have edit the light files for personal status or accomplishments.

There's plenty more to help you build a lasting, intuitive understanding of of it (just like 1, 2, 4, 8, although that sequence wasn't symmetric enough). If we imagine data storage as a light switch, we have Odd: not divisible by 2 (1, 3, 5, 7) For example, “5 mod 3 = 2” which means 2 is the remainder when you divide 5. than the number that is three more than 57(8)? 70(8). Notes on 5: One Use this strategy to add the numbers below Notes on 1: After picking 1,3,9 for ART the choices for B and F are 2,4,5,6,7,8. If 2 is 5 + 5 ≡ 10 (mod 12) After losing his previous bet, Mao bets his friend that, if the fair 6-sided die is rolled twice more. The probability of pulling out two blue socks is 3/5 × 2/4 = 3/ Alternative solution: Whether or not light B blinks with A, A will always blink alone twice before the next distance of 2 × 1/4 = 1/2 mile = 1/2 × × 1/3 = /2 = yards. Since is one more than a multiple of 4 (the number of digits in the pattern).