I had learned about Linear Feedback Shift Registers (LFSR) as a means of generating sequences of numbers. An interesting property is their ability to generate numbers that do not repeat until every distinct value has been produced exactly once. Another interesting property is that the sequence of numbers appears to be random. The sequence isn't random at all, but the illusion of randomness is convincing. The term pseudorandom is used to describe this.

A key concept of an LFSR is it generates numbers one at a time. A new number is generated each time the LFSR is invoked, so you could get a new number on some regular time interval for example.

LFSRs work with binary representations of numbers. Every number is just a series of 1s and 0s. As a result, the range of values produced by an LFSR is always a power of 2, much in the same way that powers of 10 in our decimal numbering system are common.

One small detail: an LFSR will never generate a zero, so the length of the sequence is a power of 2, minus 1. If an LFSR ever produced a zero, it would get stuck producing a stream of zeros forever. Zeros are kryptonite to LFSRs.

When I heard about the One Hertz Challenge, I got the idea to use an LFSR to generate a sequence of pseudorandom numbers every second, however, I needed a way to display the numbers to an observer.

Neopixels are RGB LEDs capable of displaying 16,777,216 distinct colors. I thought it would be fun to express a series of distinct pseudorandom numbers as a pattern of pseudorandom colors of light. Since 16,777,216 is a power of 2, an LFSR is perfectly suited for the job.

All I needed was a 1 Hertz clock, an LFSR, and a Neopixel.

Here it is in action:

I did the math. 16,777,216 seconds is a long time. If my math is correct, that's 279,620.2666 minutes, or 4,660.33777 hours, or just over 194 days.

By the way, you could do all of this in software running on an Arduino, but, why would anyone want to do it the easy way?

Here's a photo of the hardware:

The 1 Hertz Clock

The point of the challenge is to perform a task, once per second. In hardware, that means I need a clock. A clock is just a device that generates a pulse at a specific frequency.

Normally I'd turn to the good ol' 555 Timer for a job like this, but something with a cyclic period of over 194 days deserves to be precise. Besides, fake internet points are awarded for precision and I wanted an excuse to use crystals for the first time ... ever.

Crystal oscillators can be used to produce a steady, very precise pulse. They remain stable over a wide range of temperatures and are unaffected by ambient electromagnetic fields. By their nature, they vibrate at "high" frequencies, by that I mean higher than 1 Hz. For example, crystals that vibrate at 32,768 Hz are readily available; 1 Hz crystals do not exist (or they would have to be very, very, impractically large).

One way to produce a precise 1 Hz signal is to generate a precise 32,768 Hz signal, then divide that in half. Then in half again, and again and again 15 times.

32,768 = 2^15

2^15 ÷ 2^15 = 1

Dividing the frequency of a pulse is pretty easy using a cascading series of flip-flops. Flip-flops are basic building blocks of digital electronic circuits. The output of each floppy flipper is connected to the input of the next. A signal at the output of a flip-flop will pulse once for every two pulses at its input, effectively dividing the frequency by 2. Chain 15 of these together and the precise 32,768 Hz signal becomes a 1 Hz signal with no loss of precision.

Flip-flops exist as convenient integrated circuits, or "chips". For example, the 4013 is a CMOS IC that works perfectly for frequency division. But using them would require a lot of chips and a lot of wiring.

The 4060 IC contains 14 flip-flops already chained together. It also contains the oscillator circuitry that makes an external crystal do its magic. Attaching a 32,768 Hz crystal (and a capacitor...