When I was a student, lots of my fellow students approached me and asked “Where do you start to solve a computer science problem?” A question could typically be something like “write an algorithm to calculate…” I hope the following will give a good starting point and building block to get you on the road to solving the problem.Let’s work through an example. You get the question to write an algorithm or program to solve “Pascal’s Triangle”. The question might be something like “The user will enter a row number, and then the program must calculate and display the elements of that row in the Pascal Triangle”Pascal’s triangle is a geometric arrangement of the binomial coefficients in a triangle. In plain English this can be explained as “every number in the triangle is the sum of the two numbers above”. The number Zero is always invisible to the left and right of the triangle’s edges. To read more about Pascal’s Triangle, search for “Pascal’s Triangle” in Wikipedia.Here are the first few rows of Pascal’s Triangle: (The triangle starts with Row 0 and each row starts with element 0 from left to right)11 11 2 11 3 3 11 4 6 4 1Now, to solve this in an algorithm / program, there are always 4 basic ways to approach the problem. The 4 approaches are here listed from generally easiest to advanced, and also generally from worst to best.1) Hard Coding2) Iterative3) Recursive4) FunctionLets look at all 4:Hard CodingThe hard-coded solution is a silly solution! Bad programming! To solve the problem with hard coding, we will store each row as a variable/memory structure and then when required, we return that specific row. Only use this method if you are writing an exam and you are seriously running out of time!IterativeThe iterative approach is in most cases the “model answer”. This is generally a good solid solution. The solution is easy to write, and also easy for someone to understand when reading the code.We will start with hard-coding only the first row. Then we will iterate calculating through the rows until we are at the required row. For Pascal’s triangle, our algorithm will loop until the required row. Each subsequent row is calculated by using the values from the previously calculated row.RecursiveThe recursive solution is normally the “higher grade” solution. It is something that is not always understood by students. I used to call this the “job security” solution. Computer Scientists enjoy this solution.With the recursive solution we will use a function to return each element of the required row. The function will call itself to calculate the answer by calling the same function (recursively). Each time the function is called, it will return “1” if it is the first element of the row, otherwise it will return the sum of the two elements above it.FunctionThe Function is normally the best and fasted approach. It could however be very difficult to work out a formula. With a formula we want to say:f(r,e) = some formulawhere r = row number and e = element number.For Pascal’s Triangle there is a formula! nCr or n! / (r! * (n-r)!)where n = row number and r = element number.So, our algorithm will call this function with the row- and element numbers as parameters and displayed the return value.That’s it! Problem solved.

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# Of Algorithms and Modern Computer Science – Mathematics Will Never Be the Same Again!

Is it artificial intelligence or is it just a modern-day algorithm of computer science? Today, we use algorithms for all sorts of things, and they are often the backbone running many of our computer systems. And indeed as the algorithms get better, more adapted, and work better in combination – neither mathematics or human civilization will ever be the same again.If you’d like to learn more about how modern-day algorithms came about, or how they’ve changed our lives, or how the combination of algorithms and the information in digital age have come forth to change all we see and all we know – there is a very good book I’d like to recommend to you, and it is a book that I do indeed own myself. The name of the book is;”The Advent of the Algorithm – The Idea That Rules the World” by David Berlinski, published by Harcourt, New York, NY, 2000, pages 345, ISBN: 0-15-100338-6.This book was written by a mathematician and this is same author wrote the best selling mathematics book; “A Tour of the Calculus,” luckily, however this particular book is very easy to read for a layman, so I don’t have a problem recommending it to any of my friends. Most people don’t fully understand what algorithms are or how they work. But once you are done with this book you will understand, and you also understand the power behind them.Today, we use algorithms to trade stocks, search for information online, and for decision-making tools. Algorithms are used in the military, government, academia, and even by normal citizens in nearly everything we do. The corporations and businesses use them in so many ways they can’t even all be listed. Each and every day new methods, and concepts harness the value of mathematical algorithms. Indeed, I hope you will please consider owning this book for your own personal library.