## 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.

## 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.

## 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).

## 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.

## 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:
      nodeEnter.append("circle")
.attr("fill", function(d) {
var cc;
var i = parseInt(d.data);
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.