Saturday, July 23, 2016

Random Numbers in Scratch

   Every programming language has a method for generating random numbers and Scratch is no exception. 
   Load Scratch, look in the green Operators menu, and you will find a pick random block. Drag a (pick random) block into the scripting section. 

   Continually click on the block and it will report a string of numbers, 1 though 10. 
   Change the 10 to 6, click on the block several times and note that the numerical results are similar to what you would obtain by rolling a single die. 
   The pick random block actually generates pseudo-random numbers. Since every operation in every programming language has to be based on algorithms, Scratch, like every other language, must contain an algorithm (the antithesis of a random process) that generates numbers that 'look like' random numbers. Hence, the term pseudo-random numbers. A popular class of algorithms that generate pseudo-random numbers are called linear congruential generators and constitute a major topic in contemporary mathematics.
   If you are interested, request the Linear Congruential Generator.pdf, a four-page document that works through the mathematics and provides a Scratch script that implements the Park-Miller parameter set. 
   Psuedo-random numbers, hereafter called random numbers, are fun to play with and Scratch (or any other programming or scripting language) provide students the opportunity to develop dynamic solutions to many types of mathematical problems. Here's a problem in geometrical probability.
   Consider a square is divided into four smaller, congruent squares. The square in the upper right has been shaded.
  The following Scratch block, used in a Repeat block, can be used to generate 100 random points in the above square.
    How many of the 100 points would you expect to fall in the shaded square?
One could argue that since the four smaller squares are congruent, the area of the shaded square is one-fourth the area of the large square. Since the probability of picking any point is equal to the probability of picking any other point, the expectation is that 25 of the100 points, or one-fourth, would fall in the shaded area.
  As you readily see in the graphic shown below, this problem is easily modeled in Scratch. The red circular sprite, sprite 1, is used to plot each random (x, y) point.
  A yellow sprite, sprite 2, is a square with one-quarter the area of the large square (see left screenshot below). Since the yellow square is a sprite, you can click on it and move the yellow square around (see screenshot on right) within (or even outside of) the larger square. This makes the problem dynamic, as compared to the static example given at the beginning of this lecture.  
    When the program is executed (click on the green flag) 100 trials of 100 random points are computed and averaged. As you can see in the left screenshot, the average for the 100 trials is 25.27. The average for the screenshot on the right is 24.62. Both averages are close to the theoretical value of 25.
   This project can be viewed and downloaded by clicking on this link:

Sunday, July 10, 2016

Where Do Random Numbers Come From?

   The random numbers generated in a computer or calculator are not truly random. They are not produced by sampling a physical process that contains a random process, such as flipping a coin or monitoring radioactive decay.

   Computer and calculator-generated random numbers are produced by computing an algorithm. An algorithm, by its very nature, contains no random processes. Still, these computer-generated random numbers pass most statistical tests and are, for most (but not all) practical purposes, random. Random numbers produced by an algorithm are more accurately called pseudo-random numbers.           

   The most commonly used algorithm for generating psuedo-random numbers is the linear congruential generator (LCG). The defining equation for a LCG is

xn+1 = (axn + c )mod m

where xn is called the seed, a is an integer constant called the multiplier, c (also an integer constant) is called the increment, and m is the modulus.
   The project can be viewed and downloaded by clicking on this link:
    A four-page pdf document takes you through a paper and pencil exercise that explains the mathematics of a linear congruential generator and is available, free, on request. Send a request to

Wednesday, July 6, 2016

Ask Fibonacci

  During the Fall of 1998 I was researching the algorithms computer scientists use to generate random numbers (called pseudorandom numbers). One of the references I came across was this brief statement in Ivars Peterson's book, The Jungles of Randomness: A Mathematical Safari.

“At the heart of the Marsaglia-Zaman method of generating random numbers is the so-called Fibonacci sequence…"

   While waiting to see my bone doctor one morning, I remembered this statement and began to explore the patterns exhibited by the one's digit of the Fibonacci sequence. The question in my mind was, What is there about the one’s digit of the Fibonacci sequence that would interest two research mathematicians?

   Using only pencil and paper I soon made two startling discoveries (known to others but unknown to me, and probably you) about the 1's digit of any Fibonacci number. My first discovery made it possible for me to write an algorithm that computes the 1’s digit of the nth Fibonacci number.
   The project can be viewed and downloaded by clicking on this link:

Tuesday, July 5, 2016

What is iteration?

   Many of my Scratch projects are about fractals and mathematical chaos. Examples are given below. 
   Both topics are products of the computer age and are powered by a mathematical method known as iteration. Iteration is sometimes confused with looping but it is a much deeper and more mathematically powerful tool than looping.
   I have written a document that carefully, step-by-step, compares the process of graphing y = f(x) with the iteration of y = f(x). This information is basic for developing an understanding of the mathematics that underlie fractals and chaos.
   In the graphic shown below, the image on the left is the graph of the function given in the center of the graphic with the parameter c varying from 0 to 8 in increments of one.

      The image to the right is the iteration of the same function with c starting at -0.5 and varying to -2.0 in 0.005 increments

   For your free copy of this document send an email to

A Sample of My Fractal and Chaos Projects

Langton's Ant - Random Squares Experiment

Koch Snowflake

x^2+ c Plots

Chaos Game - Midpoint Formula

Mandelbrot Set [Featured Project - 4656 views]