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Why does Pascal's triangle work

Pascal's Triangle can show you how many ways heads and tails can combine. This can then show you the probability of any combination Pascal's triangle is a triangular array constructed by summing adjacent elements in preceding rows. Pascal's triangle contains the values of the binomial coefficient. It is named after the 17^\text {th} 17th century French mathematician, Blaise Pascal (1623 - 1662)

Pascal's triangle, in algebra, a triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y) n. It is named for the 17th-century French mathematician Blaise Pascal, but it is far older We write ${n \choose k},$ read 'n choose k,' for the number of different ways we can choose a subset of size $k$ from a set of $n$ elements. We have two goals: 1. Describe why the numbers in Pascal's Triangle are [.. Pascal's triangle is a never-ending equilateral triangle of numbers that follow a rule of adding the two numbers above to get the number below. Two of the sides are all 1's and because the.. Use the combinatorial numbers from Pascal's Triangle: 1, 3, 3, 1. The likelihood of flipping zero or three heads are both 12.5%, while flipping one or two heads are both 37.5%. Whew! Top 10. Gerald has taught engineering, math and science and has a doctorate in electrical engineering. Pascal's triangle shows many important mathematical concepts like the counting numbers and the..

The triangle was actually invented by the Indians and Chinese 350 years before Pascal's time. The numbers in Pascal's Triangle are the binomial coefficients of the polynomial x + 1. The triangle is used in probability to find combinations of numbers However, the fun doesn't stop here: by modifying Pascal's triangle, we can quickly calculate any number multiplied by a power of 11. For example, we could calculate 241 x 11^2. All we do is start with 2,4,1 as our first row. As we are trying to multiply by 11^2, we have to calculate a further 2 rows of Pascal's triangle from this initial row

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy Pascal's triangle and the binomial theorem mc-TY-pascal-2009-1.1 A binomial expression is the sum, or diﬀerence, of two terms. For example, x+1, 3x+2y, a− b are all binomial expressions. If we want to raise a binomial expression to a power higher than The concept of Pascal's Triangle helps us a lot in understanding the Binomial Theorem. Watch this video to know more... To watch more High School Math videos..

In Pascal's words (and with a reference to his arrangement), In every arithmetical triangle each cell is equal to the sum of all the cells of the preceding row from its column to the first, inclusive(Corollary 2) It is Pascal's Triangle that is more referred to and used. (It is a different matter that Pascal did invent a calculator to help in his father's work and this calculator was called Pascaline. This was the world's first ever digital calculator) (Please ACCEPT The relative peak intensities can be determined using successive applications of Pascal's triangle, as described above. 5. In general, spin-spin couplings are only observed between nuclei with spin-½ or spin-1. Nuclei with I > ½ (e.g. Cl, Br) have nuclear electric quadrupole moments in addition to magnetic dipole moments. The reason tha Pascal's triangle is a number pyramid in which every cell is the sum of the two cells directly above. It contains all binomial coefficients, as well as many other number sequences and patterns. , named after the French mathematician Blaise Pascal. Blaise Pascal (1623 - 1662) was a French mathematician, physicist and philosopher Why does Pascal's Triangle work? Pascal's triangle is just a part (to be more exact, a consequence) of the binomial theorem. The theorem is proved by mathematical induction (you can find that on Wikipedia or so; it's not convenient to write the proof here) and Pascal's triangle is constructed according to it

Pascal's Triangle - mathsisfun

Why does Pascal's Triangle give the powers of 11? So the first five rows are self explanatory. 1, 11, 121, 1331, 14641 are 11 0, 11 1, 11 2, 11 3 and 11 4. But then the next row is the first with double digits so it's not exactly a power of 11 anymore. It's 1 5 10 10 5 1, but then I noticed, 1 (5+1) (1+0) 0 5 1, or 161051 is indeed 11 5 In fact, if Pascal's triangle was expanded further past Row 15, you would see that the sum of the numbers of any nth row would equal to 2^n. Magic 11's. Each row represent the numbers in the powers of 11 (carrying over the digit if it is not a single number). For example, the numbers in row 4 are 1, 4, 6, 4, and 1 and 11^4 is equal to 14,641 Lesson Summary. Pascal's triangle is based on a technique known as recursion, where you can find the next number in a pattern by adding up the previous numbers. You can derive the powers of two on.

Pascal's Triangle Brilliant Math & Science Wik

1. Pascal's triangle is a triangular array of the binomial coefficients. Write a function that takes an integer value n as input and prints first n lines of the Pascal's triangle. Following are the first 6 rows of Pascal's Triangle. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5
2. The images of Pascals Triangle that I composed myself for this post were put together in Microsoft Excel, and you can download the spreadsheet if you'd like to play with the numbers yourself. Special thanks to Math Forum for the best, most simplest page on the web exploring Pascal's Triangle and the powers of 11
3. If i take the number 1.1 and raise it into those powers, i will get the same results of the Pascal's triangle. For example: 1.1^0 is equal to 1. 1.1^1 is equal to 1.1. 1.1^2 is equal to 1.21. 1.1^3 is equal to 1.331. 1.1^4 is equal to 1.4641. And so on

Pascal's triangle mathematics Britannic

Pascal's Triangle presents a formula that allows you to create the coefficients of the terms in a binomial expansion Lesson Summary. Pascal's triangle is based on a technique known as recursion, where you can find the next number in a pattern by adding up the previous numbers. You can derive the powers of two on. More rows of Pascal's triangle are listed in Appendix B. A different way to describe the triangle is to view the ﬁrst li ne is an inﬁnite sequence of zeros except for a single 1. To obtain successive lines, add every adjacent pair of numbers and write the sum between and below them. The non-zero part is Pascal's triangle Pascal's Triangle. Blaise Pascal (1623-1662) is associated with the triangle of numbers which bears his name, although it is known as Tartaglio's Triangle in Italy, and was known at least 700 years before Pascal by Indian, Chinese, and other mathematicians, perhaps a long time before that too. Our interest here is with the Binomial Theorem Pascal's Triangle is a triangular array of numbers where each number on the interior of the triangle is the sum of the two numbers directly above it. It was named after French mathematician Blaise Pascal. There are many interesting things about the Pascal's triangle. In this post, we explore seven of these properties

Why does Pascal's triangle work to find coefficients in

Pascal's Triangle The relative intensitites of the lines in a coupling pattern is given by a binomial expansion or more conviently by Pascal's triangle. To derive Pascal's triangle, start at the apex, and generate each lower row by creating each number by adding the two numbers above and to either side in the row above together Introduction. The Pascal triangle is a sequence of natural numbers arranged in tabular form according to a formation rule. Here's an example for a triangle with 9 lines, where the rows and columns have been numbered (zero-based) for ease of understanding Now, by considering the different powers of ������ and ������ and using Pascal's triangle, work out the coefficients of the expansion (2 ������ − 2 ������) . Answer . Part 1. Recall that we can write out the rows of Pascal's triangle by pairwise adding the terms in the previous rows The choice of k things from a set of n things without replacement and where order does not matter is called a combination. Examples: 1. Picking three team members from a group. 2. Picking two deserts from a tray. 8.2 Pascal's Triangle Motivational Problem Calculation of Combinations: Consider a grid that has 5 rows of 5 square An infinite multiplication of fractions less than or equal to 1 must converge to 0. So, in an infinite Pascal's triangle, the middle term, although it increases without bound, is 0% of the row's total.---The above paragraph is incorrect. A lot more work must be done to show that the product of the remaining fractions goes to zero

Pascal's triangle is a table of numbers in the shape of an equilateral triangle, where the k-th number in the n-th row tells you how many combinations of k elements there are from a set of n elements (Note that we follow the convention that the top row, the one with the single 1, is considered to be row zero, while the first number in a row. Pascal's theorem is a very useful theorem in Olympiad geometry to prove the collinearity of three intersections among six points on a circle. The theorem states as follows: There are many different ways to prove this theorem, but an easy way is to use Menelaus' theorem And here comes Pascal's triangle. It tells you the coefficients of the progressive terms in the expansions. For example, the first line of the triangle is a simple 1. And indeed, (a + b)0 = 1. The second line is 1 1. And in fact, (a +b)1 = 1a +1b. The third line is 1 2 1 Pascal's triangle has many unusual properties and a variety of uses: Horizontal rows add to powers of 2 (i.e., 1, 2, 4, 8, 16, etc.) The horizontal rows represent powers of 11 (1, 11, 121, 1331, 14641) for the first 5 rows, in which the numbers have only a single digit Well, we can work this out now using the Binomial Theorem together with Pascal's Triangle. With n = 2, x = a , and y = b in the formula, we find: The values of the binomial coefficients can be found using a Calculator (using the n C r function), or by Pascal's Triangle

The Hockey Stick property and the less well-known Parallelogram property are two characteristics of Pascal's triangle that are both intriguing but relatively easy to prove. This article explains what these properties are and gives an explanation of why they will always work Rewrite the table as a triangle, and look at how each number is. related to the two above it: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1. Student: Cool! The next row should be 1, 6, 15, 20, 15, 6, 1 -- you just add the two above. Mentor: Exactly. This is Pascal's Triangle

Properties of Pascal's Triangle Live Scienc

• The (n+1) Rule, an empirical rule used to predict the multiplicity and, in conjunction with Pascal's triangle, splitting pattern of peaks in 1 H and 13 C NMR spectra, states that if a given nucleus is coupled (see spin coupling) to n number of nuclei that are equivalent (see equivalent ligands), the multiplicity of the peak is n+1. eg. 1: The.
• Given an integer numRows, return the first numRows of Pascal's triangle. In Pascal's triangle , each number is the sum of the two numbers directly above it as shown: Example 1
• g computations in problems involving probability and statistics, it's often helpful to have the binomial coefficients found in Pascal's triangle. These numbers are the results of finding combinations of n things taken k at a time. For quick reference, the first ten rows of the triangle are shown
• ent 17th Century scientist, philosopher and mathematician. Like so many great mathematicians, he was a child prodigy and pursued many different avenues of intellectual endeavour throughout his life. Much of his early work was in the area [
• Pascal's wager, Practical argument for belief in God formulated by Blaise Pascal.In his Pensées (1657-58), Pascal posed the following argument to show that belief in the Christian religion is rational: If the Christian God does not exist, the agnostic loses little by believing in him and gains correspondingly little by not believing. If the Christian God does exist, the agnostic gains.
• Once students are ready to begin work on the coloring problems (project problems number 3 or 4), distribute several copies of the Pascal's triangle coloring sheets for them to use. You might leave questions about alternatives to generating the values in Pascal (rather than the remainders) until this activity, since the coloring may lead some. Top 10 Secrets of Pascal's Triangle by Brett Berry

Pascal discovered new properties of the triangle and solved problems using it, therefore the triangle became known as Pascal's Triangle. The dice problem asks how many times one must throw a pair of dice before one expects a double six while the problem of points asks how to divide the stakes if a game of dice is incomplete An interesting property of Pascal's Triangle is that its diagonals sum to the Fibonacci sequence, as shown in the picture below: It will be shown that the sum of the entries in the n -th diagonal of Pascal's triangle is equal to the n -th Fibonacci number for all positive integers n . Suppose = sum of the n -th diagonal and is the n- th. The disadvantage in using Pascal's triangle is that we must compute all the preceding rows of the triangle to obtain the row needed for the expansion. The following method avoids this. It also enables us to find a specific term — say, the 8th term — without computing all the other terms of the expansion

A proof by example, i.e. this pattern holds in the small portion of Pascal's Triangle that I have drawn, is not a proof period, combinatorially or otherwise. The general case of such a property could be verified combinatorially, but simply observing it would not constitute a combinatorial proof in itself. At least that's the way I see it These, of course, are the lines from Pascal's Triangle. And yes, it does work for all positive whole number values of the index. Prove to yourself by algebra that, (1 + x) 4 = 1 + 4x + 6x 2 + 4x 3 + x 4. In fact there is a general rule that (a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b

The Relationship Between Pascal's Triangle & Combinations

It is in this work that we find what is known as Pascal's Wager. The gist of the Wager is that, according to Pascal, one cannot come to the knowledge of God's existence through reason alone, so the wise thing to do is to live your life as if God does exist because such a life has everything to gain and nothing to lose The work is now lost but Leibniz and Tschirnhaus made notes from it and it is through these notes that a fairly complete picture of the work is now possible. Although Pascal was not the first to study the Pascal triangle, his work on the topic in Treatise on the Arithmetical Triangle was the most important on this topic and, through the work of.

What Is Pascal's Triangle Used for Today

Now it is easy to work out how many dots: just multiply n by n+1. Dots in rectangle = n(n+1) But remember we doubled the number of dots, so . Dots in triangle = n(n+1)/2. We can use x n to mean dots in triangle n, so we get the rule: Rule: x n = n(n+1)/ Pascal is a major character in Disney's 2010 animated feature film, Tangled. He is Rapunzel's pet chameleon, with the ability to change colors as both a means of camouflage and expression. With a feisty personality, Pascal acts as Rapunzel's confidant and self-appointed protector. 1 Background..

Video: Pascal's Triangle and Powers of 11 - Articles Factor

RELATED: What is Pascal's Triangle? Interior and exterior angles. Before we get too far into our story about triangles and the total number of degrees in their three angles, there's one little bit of geometric vocabulary that we should talk about. And that is the difference between an interior and an exterior angle Due to the definition of Pascal's Triangle, . History. Pascal's identity was probably first derived by Blaise Pascal, a 17th century French mathematician, whom the theorem is named after. Pascal also did extensive other work on combinatorics, including work on Pascal's triangle, which bears his name. He discovered many patterns in this triangle. Blaise Pascal continued his mathematical work, inventing a mechanical calculator later called the pascaline at the age of 18. Pascal's later mathematical work dealt with geometry and probability. In addition to his work in mathematics, Pascal contributed to the scientific fields of hydrodynamics and hydrostatics in his twenties Micrograph of vasculature in the adult zebrafish brain. Jessica Plavicki, an assistant scientist in the Pharmacuetical Sciences Division, was among only 11 winners of the 2015 Cool Science Image Contest.. Her entry of the micrograph of vasculature in the adult zebrafish brain was captured with a confocal microscope And you don't need to give up the power, the Pascal language is as powerful as you want it. No Makefiles Unlike most programming languages, Pascal does not need Makefiles. You can save huge amounts of time, the compiler just figures out itself which files need to be recompiled. Pascal compilers are Fast with a big F and Free Pascal is no exception Pascal's triangle - Wikipedi

Pascal's principle, an experimentally verified fact, is what makes pressure so important in fluids. Since a change in pressure is transmitted undiminished in an enclosed fluid, we often know more about pressure than other physical quantities in fluids. Moreover, Pascal's principle implies that the total pressure in a fluid is the sum of the. The Binomial theorem tells us how to expand expressions of the form (a+b)ⁿ, for example, (x+y)⁷. The larger the power is, the harder it is to expand expressions like this directly. But with the Binomial theorem, the process is relatively fast! Created by Sal Khan. This is the currently selected item Binomial Coefficients in Pascal's Triangle. Numbers written in any of the ways shown below. Each notation is read aloud n choose r.These numbers, called binomial coefficients because they are used in the binomial theorem, refer to specific addresses in Pascal's triangle.They refer to the nth row, rth element in Pascal's triangle as shown below Consider Pascal's triangle. When Pascal was consulted by a gambler about the odds of the outcomes of a die and the nature of stakes, he invented the theory of probability to solve these problems. Pascal's triangle is a neat triangle formed by binomial coefficients. The triangle acts as a table that one refers to while expanding the binomial. Pascal's Triangle Recall Pascal's Triangle, in which the outer edges are filled with ones and each inner element is the sum of its upper adjecent elements. This is a fascinating creature; Pascal's Triangle has always seemed to pop up in the strangest places: number theory, probability theory, even polynomial expansion An early version of Pascal's Calculator or Pascaline #3 He did important work concerning atmospheric pressure and vacuum. In 1643, Italian physicist Evangelista Torricelli filled a glass tube with mercury and inverted it into a column of mercury thus inventing the barometer, which is used to measure atmospheric pressure.Torricelli surmised that the space at the top of the tube was a vacuum In addition, Pascal's Wager does not prove the existence of G-d, but the logic in believing or striving to believe in him. Pascal's Wager puts the choice and onus on the individual, whereas, the Atheist Wager places the onus on G-d without a clear foundation

Pascal triangle pattern is an expansion of an array of binomial coefficients. Each number in a pascal triangle is the sum of two numbers diagonally above it. Just copy and paste the below code to your webpage where you want to display this calculator. Generating a Pascals Triangle Pattern is made easier with this online calculator Pascal - Pointer arithmetic. There are four arithmetic operators that can be used on pointers: increment,decrement, +, -. 2. Pascal - Array of pointers. You can define arrays to hold a number of pointers. 3. Pascal - Pointer to pointer. Pascal allows you to have pointer on a pointer and so on. 4 However, for quite some time Pascal's Triangle had been well known as a way to expand binomials (Ironically enough, Pascal of the 17th century was not the first person to know about Pascal's triangle) Binomial Theorem Calculator. A closer look at the Binomial Theorem. The easiest way to understand the binomial theorem is to first just look at. Over the years, Pascal's constant work took a further toll on his already fragile health. Pascal died of a malignant stomach tumor at his sister Gilberte's home in Paris on August 19, 1662

Consider again Pascal's Triangle in which each number is obtained as the sum of the two neighboring numbers in the preceding row. The way the entries are constructed in the table give rise to Pascal's Formula: Theorem 6.6.1 Pascal's Formula top Let n and r be positive integers and suppose r £ n. Then. Example 6.6.5 Deriving New Formulas from. Pascal's Triangle is defined such that the number in row and column is . For this reason, convention holds that both row numbers and column numbers start with 0. Thus, the apex of the triangle is row 0, and the first number in each row is column 0. As an example, the number in row 4, column 2 is . Pascal's Triangle thus can serve as a look-up. Pascal's response is to invent an entirely new branch of mathematics, the theory of probability. This theory has grown over the years into a vital 20th century tool for science and social science. Pascal's work leans heavily on a collection of numbers now called Pascal's Triangle, and represented like this

As far as we know, this is the only page on the web showing this formula and how it fits with Pascal's triangle and that's why this page has a little copyright note at the bottom. We call it THE UNKNOWN FORMULA and it's now featured in The Perfect Sausage and Other Fundamental Formulas Write out the first few powers of 11. Do they remind you of Pascal's triangle? Why? Why does the Pascal's triangle pattern break down after the first few powers? (Hint: consider (a+b) m where a=10 and b=1). To finish, let's return to a human family tree. Suppose that the probability of each child being male is exactly 0.5 so that exactly half.

Pascal's Triangle is a simple, but powerful triangle formed by creating a triangle with three 1's to begin. For each row thereafter you simply write 1's on both ends, and find the middle. Pascal's Triangle: Each number in the triangle is the sum of the two directly above it. The rows of Pascal's triangle are numbered, starting with row $n = 0$ at the top. The entries in each row are numbered from the left beginning with $k = 0$ and are usually staggered relative to the numbers in the adjacent rows

1. Pascal's Principle Practice. Pascal's Principle When force is applied to a confined liquid, the change in pressure is transmitted equally to all parts of the fluid. Draw a bottle of water with arrows to illustrate the regular exerted pressure. Then draw a water bottle that you squeeze
2. In math class, there are 24 students. The teacher picks 4 students to help do a demonstration. How many different groups of 4 could she have chosen? In how many ways can 10 people wait in line for concert tickets if Pete is last in line? The teacher has listed 10 short stories and 6 books on the course syllabus
3. Pascal-K1N6 will convert each stack of 250 Spare Parts, into S.P.A.R.E. Crate which do stack. To convert a S.P.A.R.E. Crate back into 250 Spare Parts simply click. Pascal-K1N6 is found to the West of Rustbolt at Mechagon at 71.2 32.7 Coordinates for TomTom users: /way Mechagon 71.31 32.40 Pascal-K1N6 Parts into Crate
4. The triangle below was generated from iterations by trisecting the line segments that make up the largest triangle. Does this new iteration correlate to the pattern found in Pascal's triangle mod 3? See how this compares to Pascal's Triangle in mod 3!. View the GSP construction and tool for this figure, and adjust the number of iterations shown
5. Blaise Pascal and Pierre de Fermat invented probability theory in 1654 to solve a gambling problem related to expected outcomes. An intellectual friend of Pascal's wanted to figure out the best time to bet on a dice game, and how to fairly divide the stakes if the game was stopped midway through. He asked his math genius friend Pascal for help
6. Pascal's Wager is the name given to an argument due to Blaise Pascal for believing, or for at least taking steps to believe, in God. The name is somewhat misleading, for in a single section of his Pensées, Pascal apparently presents at least three such arguments, each of which might be called a 'wager'—it is only the final of these that is traditionally referred to as Pascal.
7. Why do the numbers from Pascal's Triangle (which is a visual array created by Blaise Pascal circa 1623 to 1662) become the coefficients of each term for our expansion? The numbers in each row of Pascal's Triangle are created by adding the closest two terms of the preceding row

Pascal's Triangle and the Binomial Theorem Don't

1. g language, as well as an apologetic argument. Even more, Pascal is the inventor of the first calculator and the first public transportation system. Blaise pascal really did make a great impact to the world and of course, to myself
2. Figure 4.1: Pascal's Triangle Pascal's triangle can be used to compute the likelihood of any event with even odds occurring. Consider, for instance, the odds that a couple expecting their first child will have a boy; the answer, with even odds, is one-half and is in the second line of Pascal's triangle
3. As you might imagine, drawing Pascal's Triangle every time you have to expand a binomial would be a rather long process, especially if the binomial has a large exponent on it. People have done a lot of studies on Pascal's Triangle, but in practical terms, it's probably best to just use your calculator to find n C r , rather than using the Triangle
4. Pascal's Traité du triangle arithmétique (Treatise on Arithmetical Triangle) was published posthumously in 1665. Blaise Pascal's version of the triangle. — Source: Wikipedia. Pascal's triangle has many amazing properties and contains many special patterns of numbers. Below we are presenting few of the most incredible examples

Patterns in Pascal's Triangle - Alexander Bogomoln

1. Thoughts spoke on and defended the Christian faith, and is considered to be Pascal's most influential theological work. It was incomplete at the time of his death, and was published posthumously. It is also considered to be a landmark in the world of French prose. 9. Blaise Pascal invented the first form of modern public transportation in Pari
2. Pascal's Triangle mod 2 with highlighted matching regions. Theorem: For the mod 2 Pascal's triangle, each new block of rows from row through row -1 has exactly two copies of the first rows (rows 0 through -1) with a triangle of 0's in between. Proof: We will prove the claim inductivel
3. See below. The Fibonacci sequence is related to Pascal's triangle in that the sum of the diagonals of Pascal's triangle are equal to the corresponding Fibonacci sequence term. This relationship is brought up in this DONG video. Skip to 5:34 if you just want to see the relationship
4. Pascal's work on probability theory is widely known due to his correspodence with Fermat. (Renyi) It was in this year that he published Traite du triangle arithmetique . After another religious conversion in 1654, in which Pascal fully commit himself to God, his writings were primarily of a philosophical nature

An if-then statement can be followed by an optional else statement, which executes when the Boolean expression is false.. Syntax. Syntax for the if-then-else statement is −. if condition then S1 else S2; Where, S1 and S2 are different statements.Please note that the statement S1 is not followed by a semicolon Pascal's life is inseparable from his work.—A. J. Krailsheimer. Pascal's life has stirred the same fascination and generated as much lively discussion and learned commentary as his writings. This is largely attributable to his intriguing, enigmatic personality Pascal was originally absent in New Horizons, but he returned as part of the Summer update on July 3rd.He can be spotted when diving in the ocean for scallops, where he may offer the player to trade the scallop for DIY recipes of the Mermaid set, or pearls required to craft the Mermaid Series furniture.He can also give the player mermaid-themed dresses and matching accessories In Pascal's triangle, the sum of all the numbers of a row is twice the sum of all the numbers of the previous row. So, the sum of 2nd row is 1+1= 2, and that of 1st is 1. Again, the sum of 3rd row is 1+2+1 =4, and that of 2nd row is 1+1 =2, and so on. This major property is utilized here in Pascal's triangle algorithm and flowchart