Illustrative Mathematics 3.OA Patterns in the multiplication table
... This task is intended for instruction. The goal is to look for structure and identify patterns and then try to find the mathematical explanation for this. This problem examines the ''checkerboard'' pattern of even and odd numbers in a single digit multiplication table. The even numbers in the table ...
... This task is intended for instruction. The goal is to look for structure and identify patterns and then try to find the mathematical explanation for this. This problem examines the ''checkerboard'' pattern of even and odd numbers in a single digit multiplication table. The even numbers in the table ...
The quadratic recurrence for matchings of the 2-by
... On the other hand, the Fibonacci sequence F_n, extended backwards, has F_n an integer for all integers n, so it’s desirable to find reciprocity theorems that cover cases like this. It’s desirable to extend this to other functions f() that satisfy a linear recurrence equation with constant coefficien ...
... On the other hand, the Fibonacci sequence F_n, extended backwards, has F_n an integer for all integers n, so it’s desirable to find reciprocity theorems that cover cases like this. It’s desirable to extend this to other functions f() that satisfy a linear recurrence equation with constant coefficien ...
Indexed Classes of Sets Let I be any nonempty set, and let S be a
... A function is said to be recursively defined if the function definition refers to itself. In order for the definition not to be circular, the function definition must have the following two properties: (1) There must be certain arguments, called base values, for which the function does not refer to itse ...
... A function is said to be recursively defined if the function definition refers to itself. In order for the definition not to be circular, the function definition must have the following two properties: (1) There must be certain arguments, called base values, for which the function does not refer to itse ...
Solutions
... (a) Define a sequence a1, a2, etc. such that ai equals the maximum height of the ball in between bounce number i and bounce number i+1. This sequence would be a geometric one, because each bounce reduces the height of the ball by a constant ratio. To solve this part, we must ascertain the value of a ...
... (a) Define a sequence a1, a2, etc. such that ai equals the maximum height of the ball in between bounce number i and bounce number i+1. This sequence would be a geometric one, because each bounce reduces the height of the ball by a constant ratio. To solve this part, we must ascertain the value of a ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.