Here’s something interesting : work out the cubes of 1, of 5 and of 3, then add them together.
You should get 153.
Now, try adding 16 cubes, 50 cubed and 33 cubed. Lo and behold, you get 165033!
It doesn’t stop there… try adding 166 cubed, 500 cubed and 333 cubed. Can you guess the answer before you work it out?
Why does this work?
Sometimes, when mathematicians are presented with little facts like this, they dismiss them. “After all,” the mathematician might say. “This only works because we write numbers in base 10” (that is, with ten different digits). “If we used a different base, it wouldn’t work. This is not a property of the numbers themselves, but also of the base we write them in.
However, if you work out the algebra behind why this works, you realize it’s not just base 10. In fact, if b is any base that’s two less than a multiple of six, then the sum of the cubes of (b-4)/6, b/2 and (b-1)/3 is special.
There are lots of numbers that are two less than a multiple of six. Sixteen is one, and that’s a number base that’s often used in computing. So, there’s another sequence of sums of cubes that is unremarkable in base 10, but looks amazing in base 16.
In base 10, the cubes of 2, 8 and 5 add up to 645. Nothing exciting there, until we convert the numbers to base 16 – then the answer reads 285 instead of 645. In the standard computer science way of writing base 16 numbers, I’d say “add the cubes of 0x2, 0x8 and 0x5, and you get 0x285”.
As in base 10, this is the start of an infinite chain of amazing base 16 arithmetic facts : 0x2A cubed plus 0x80 cubed plus 0x55 cubed is 0x2A8055 (42 cubed plus 128 cubed plus 85 cubed is 2785365), then 0x2AA cubed plus 0x800 cubed plus 0x555 cubed is 0x2AA800555 (I’ll let you check this one), and so on.
And there’d be other chains of sums of cubes in base 22, base 28, base 34 and so on, each chain looking amazing in its respective base, yet quite ordinary in any other base.
Hat tip : I saw the first of these amazing chains (the base 10 one) in Professor Stewart’s Hoard of Mathematical Treasures. Do check it out, it’s full of amazing tidbits like this one.
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