Both France and Germany were caught up in the age of revolution which swept Europe in the late 18th Century, but the two countries treated mathematics quite differently.
Time Life Books Gullberg, Jan. Hailstone Numbers Choose a positive integer. If your number is even, divide it by two. If your number is odd, multiply it by three and add 1.
Take your new number as the starting number, and repeat until you can't go any farther. The process sounds easy enough, but how to predict what will happen may be less obvious.
Hailstone numbers are generated by the simple mathematical process described above. A small integer value of n can demonstrate the process.
To compare hailstone sequences from different starting numbers, it is useful to note both the problem boolean algebra value attained, and the number of computations needed to reach 1.
To compare, change n by a small amount and note what happens to the sequence.
Here the largest value is again 16, but the number of steps is 8. How high will hailstone numbers go? And just how many steps are needed? Some larger examples may show greater variation attained in hailstone number sequences. The pattern is 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
Here the maximum value is 88, and the length is This is where the surprises of hailstone numbers become more obvious. The sequence is 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. The maximum value is 40, and the number of computations only The sequence has a maximum value of 9, and takes computations to reach 1!
So far, it appears that the maximum values and numbers of computations needed are in some way related to the size of the starting number.
Perhaps more notable is the contrast between even and odd starting numbers. There is, as yet, no complete understanding of the apparently chaotic nature of the sequences produced by this simple procedure.
Also, another puzzling question remains.
Will any starting number used for a hailstone sequence eventually reach 1? From the examples above, it would appear so. Certainly the hailstones these numbers are named for have somewhat predictable behavior.
Hailstones go up and down inside of clouds, gaining size and mass with each pass through.Have your students explain the entire process they used in simplifying the gate circuit: developing the Boolean expression, simplifying that expression using Boolean algebra techniques, and then developing a new gate circuit from the simplified Boolean expression.
Problem 2: Karnaugh Maps with “Don’t Cares” Karnaugh Maps are useful for finding minimal implementations of Boolean expressions with only a few variables. This introduction to Boolean algebra explores the subject on a level accessible even to those with a modest background in mathematics.
The first chapter presents the algebra of sets from an intuitive point of view, followed by a formal presentation in chapter two of Boolean algebra as an abstract algebraic system, with no reference to applications.
Papers (publications and preprints): Darij Grinberg, Peter Olver, [Prop] The n body matrix and its determinant, arXiv The primary purpose of this note is to prove two recent conjectures concerning the n body matrix that arises in recent papers of Escobar--Ruiz, Miller, and Turbiner on the classical and quantum n body problem in d-dimensional space.
Based on the rules and theorems of boolean algebra, we can simplify the equations of a boolean problem. The logic for the simplification always follows the truth table as, when we have two false values the result will always .
Boolean algebra is the algebra of two values - true and false. For example, the statement "6 is an even number" is true, and the statement "The Neva flows into the Caspian Sea" is false. These 2 elements - true and false - are called constants and they are assigned, respectively, logical 1 and logical 0.