The Collatz Conjecture

The Collatz conjecture is an unsolved problem that dates back to 1937, though mathematicians have proven it for certain ranges.

It revolves around a piecewise function:

$f(n)=3n+1$ for odd values of n

$f(n)=n/2$ for even values of n

This function takes a natural number (which can be any integer greater than zero) as its domain. The conjecture states that for all $n>1$, the sequence created by recursively calling the function will converge to $1$, at which point it becomes a loop.

In other words, if you take the value of the previous function call and put it through $f(n)$ enough times, $f(n)$ will eventually equal $1$.

Visualization

I visualized the Collatz conjecture using HTML5 Canvas. Each bar represents a "step" in the Collatz sequence starting at the inputted number.

The visualization renders bars until $f(n)$ reaches $1$. You can see in slow motion how a sequence jumps around before eventually falling to zero.

Note that due to limitations of the canvas, some long sequences may not render correctly.

Interesting Sequences

Some sequences have interesting results due to the properties of the function. Try starting with some of the following:

• Powers of 2
• Prime numbers
• Large numbers
• Doubling your previous input
• Odd vs. even starting numbers

Instructions

The box labeled "Starting value" sets the starting point for the conjecture, i.e. the first number in the Collatz sequence.

Enter a value you wish to visualize and click "See it!" to begin the animation.

Use the speed slider to control how fast the animation moves.