A neat multi-base identity

I’m about to start teaching my one-week intensive math workshop for middle and high school students, which I call “Numbers in the Summer”. I hope to report a few fun tidbits from that workshop, for which I use materials from the awesome Park City Math Institute. Here’s a very quick one. Note that in base 10, we have 10^2 + 11^2 + 110^2 = 111^2. If you try it in base 2 (binary), you’ll find it also works! So here’s the question: for which bases is this equation true? Try it out before you click for more…

All right, hopefully you tried base 3, or 5, or 16, or something else. If you did, you’ll have seen that it worked in whatever base you tried—because it works in any base! Here’s the algebra to prove it, in some arbitrary base b: the terms on the left are 10_b = b,\:\: 11_b = b+1,\:\: 110_b = b^2 + b. Hence the total on the left is b^2 + (b+1)^2 + (b^2 + b )^2, which expands to b^4 + 2b^3 + 3b^2 + 2b + 1, which is also the expansion of (b^2 + b + 1)^2 = (111_b)^2. Fun, yes? More to come soon, I hope.

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