Friday, June 02, 2023

A Chiral Aperiodic Monotile

An update to the aperiodic tile. This update shows that a shape that tiles the plane aperiodically without reflections is possible. The original one did not.

Click here for the article.

Saturday, April 01, 2023

Mathematicians have finally discovered an elusive ‘einstein’ tile

A 13-sided shape known as “the hat” has mathematicians tipping their caps.

It’s the first true example of an “einstein,” a single shape that forms a special tiling of a plane: Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats.

Click here for more information.

Sunday, July 31, 2022

Value-Counting Up to N

Some interesting properties arise when value-counting the integers sequentially up to N using N digits or fingers and comparing the number of values to the prime-exact equation; with a simple method for testing primes and prime powers (particularly Mersenne and Fermat primes).

Click here to read my paper.

Wednesday, December 22, 2021

Primality Testing and Factoring Using Pascal's Triangle

An interesting if not impractical way of primality testing and factoring a number using Pascal’s Triangle.

Click here to read my paper.

Sunday, October 13, 2019

Collatz Conjecture

I've previously linked to Jason Davies website for another article. He has another JavaScript program for the Collatz Conjecture.

To pretty it up, remove the circle fill in collatz.css and modify the circle append (line 83) in collatz.js as follows:
          .attr("fill", function(d) {
            var cc;
            var i = parseInt(;
            if ((i && (i & (i - 1)) === 0) && (i <= 16)) {
              cc = "#0000ff";
            else if ((i % 3) === 0) {
              cc = "#c8c8c8";
            else if ((i % 6) === 1) {
              cc = "#ffff00";
            else if (((i % 2) === 1) || (((i % 3) === 2) && (((i / 2) % 2) === 0))) {
              cc = "#ffa500";
            else {
              cc = "#000000";
            return cc;
         .attr("r", 5);
I was only interested in the initial node for those in orange and yellow.

Thursday, October 10, 2019

Minimal Set for Powers of 2

The minimal set for powers of 2 is currently nondeterministic and can be shown to be more complex than previously proposed.

Click here for my analysis on it.

Monday, July 01, 2019

Mathematicians Discover the Perfect Way to Multiply

Four thousand years ago, the Babylonians invented multiplication. Last month, mathematicians perfected it.

On March 18, two researchers described the fastest method ever discovered for multiplying two very large numbers. The paper marks the culmination of a long-running search to find the most efficient procedure for performing one of the most basic operations in math.

“Everybody thinks basically that the method you learn in school is the best one, but in fact it’s an active area of research,” said Joris van der Hoeven, a mathematician at the French National Center for Scientific Research and one of the co-authors.

Click here and here for more information.