Sometimes it's nice to have something to procrastinate with that still feels like somewhat productive work; as a math nerd who enjoys all sorts of recreational math puzzles, I had to throw my hat into the ring and share my attempt at a solution manual for the Four Fours challenge, which is an ongoing project of mine.
The premise is straightforward: find the simplest mathematical expression for every whole number from 0 to some maximum, using only common mathematical functions/symbols and the digit 4. However, when you increase the maximum past ~100, the challenge grows considerably harder, and it is necessary to expand and re-evaluate what counts as "common" notation.
This puzzle is over a century old, and several other webpages have discussed it and posted their own solutions with different interpretations of the prompt and ranges of notation:
Wikipedia: https://en.wikipedia.org/wiki/Four_fours
Numberphile: https://www.youtube.com/watch?v=Noo4lN-vSvw
Wheels.org: https://www.wheels.org/math/44s.html
University of Toronto: https://www.math.toronto.edu/mathnet/questionCorner/fourfours.html
Pat Ballew: https://pballew.blogspot.com/2018/12/before-there-were-four-fours-there-were.html
Four Fours: https://sites.google.com/view/four-fours/
EyeGate: https://web.archive.org/web/20110802104248/http://eyegate.com/showgal.php?id=7
There are also a few publications in mathematical journals which discuss the history of the problem, but they are unfortunately paywalled and not publicly accessible. Two websites in particular provide very detailed lists of solutions, each developing their own (very different) philosophies on what notation is acceptable for the problem:
Paul Bourke: https://paulbourke.net/fun/4444/
David Wheeler: https://dwheeler.com/fourfours/
In general, I adopt much of the notation used in these two solution manuals, with the exception of unconventional repeating decimals, the square() function, and bitwise logical operators. I also make use of trigonometric expressions which I have not seen used much in previous discussions of the problem. My solutions for the integers 0 through 1000 are presented here:
If you enjoy these types of puzzles I would encourage you to give it a try! The beauty of Four Fours is that you can create your own rules for "allowed" notation, and either stop at some maximum number or see how high you can reach. (And if you happen to find a solution for a particularly difficult number which is simpler and/or more elegant than mine, you're more than welcome to reach out to me and I can include your solution with a credit if you want 😉)