These Simple Equations Are Levels of an Infinite Pattern

You probably know 2+2 = 2x2, and you might know 1+2+3 = 1x2x3, but did you know those equations are connected and are two levels within an infinite family of similar equations? Let me show you some crazy facts, graphs, and mysteries that are hiding within this cool number pattern!

I only listed “solutions” for up to "level" 11, so leave a comment if you figure out the solutions for future levels, or if you figure out any simple proof for how one of the layer’s solutions are the only possible solutions of that layer. It’s a fun challenge for the brain! The main “question/puzzle” of this episode doesn’t have an official name (it’s an underrated concept) but can be described as “which multisets of positive integers have an equal sum and product?” (or replace "positive integers" with "numbers" in that if you want to include the infinite spectrum of non-integer solutions).

I made the graphs that are in this episode on the website Desmos (a useful free graphing calculator). Note that the non-integer coordinate numbers they are showing are approximations, and many of them would have infinite decimals. Also, part of the soundtrack in this episode is a beat I made on a livestream from scratch, to show how I make my soundtrack beats. You can see that on my bonus ‪@Domotro‬ channel here:    • Mathematically Making a Soundtrack Be...  

Check out my Patreon at   / comboclass   if you want to support the channel, get some cool bonus content (like behind-the-scenes clips, my music, etc) and get your name on a whiteboard in the Grade -1 finale! Thanks a lot to all of my supporters there: Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, Joseph Rissler, Plenty W, Quinn Moyer, Julius 420, Philip Rogers, Ilmori Fajt, Brandon, August Taub, Ira Sanborn, Matthew Chudleigh, Cornelis Van Der Bent, Craig Butz, Mark S, Thorbjorn M H, Mathias Ermatinger, Edward Clarke, and Christopher Masto, Joshua S, Max, Joost Doesberg, Adam, Chris Reisenbichler, Stan Seibert, Amy, Izeck, and Beugul.

If you use Discord, join us on the Combo Class channel where there are lots of fun conversations going on:   / discord  
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And make sure you're also subscribed to my other channel ‪@Domotro‬ for my shorts, livestreams, and bonus videos.

By the way, if you happen to have any broken clocks (or spare dice) and want them in the background of the Combo Classroom, you're in luck. I'll be starting a PO Box later this month for people to send me stuff like that.

[DISCLAIMER: DO NOT TRY TO COPY ANY OF THE PARTS OF THIS EPISODE INVOLVING FIRE. In general, never copy any physical actions you see in Combo Class, such as anything involving fire, tools, or other science experiments.]