Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics". Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems." He also offered US$500 for its solution. The sequence of numbers involved is sometimes referred to as the hailstone sequence, hailstone numbers or hailstone numerals (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.
It is also known as the 3 n + 1 problem, the 3 n + 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. It is named after mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence.
If the previous term is odd, the next term is 3 times the previous term plus 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. The Collatz conjecture is one of the most famous unsolved problems in mathematics. The Collatz conjecture states that all paths eventually lead to 1. You can also choose your own colors by coloring a black and white chart.įeel free to see our Tricks for learning your Times tables Chart.Directed graph showing the orbits of small numbers under the Collatz map, skipping even numbers. Colorful or black and white Charts: A colorful chart will help you remember more easily.Complete Charts or Blank Charts to be filled in: Complete charts are for learning the multiplication tables (or as a memory aid), while blank chart to be filled in are ideal for practice the Times tables.Different sizes of charts: Chart from 1 to 10 (10x10), Chart from 1 to 12 (12x12), Chart from 1 to 15 (15x15) or Chart from 1 to 20 (20x20).Once you have printed the Chart, you can post it on the wall near your desk or put it into your school bag. They are in A4 standard format, available as free PDF files to download.
Multiplication Charts, also called Times tables Charts, are essential tools to learn the multiplication tables! To help you in this task, we offer you many Printable multiplication charts.
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