Why four is the nemesis of happy numbers


Sometimes when I read about math I get the sense that mathematicians are able to accurately construct extraordinarily accurate pictures that neither language nor intuition can possibly touch. That kind of math can predict the movement of photons, the shape of the universe, the interactions of multiple dimensions, and the drift of galaxies.


Other times when I read about math I get the sense that mathematicians are coming up to me with a sheet of construction paper that has three lop-sided circles and smiley face on it, and saying "I made this!" Happy numbers, and their nemesis, the spiral of four, fit into that latter category of math. I suppose they're an interesting phenomenon, but neither I, nor as far as I can tell any mathematician, have found them to be anything more than a quirk.

Finding happy numbers is simple. Pick a number. Separate out its digits. Square them. Add them together to form a new number. Repeat. If the eventual product is one, you've got a happy number. For example:


8, 2

64 + 4 = 68

6, 8

36 + 64 = 100

1, 0, 0

1 + 0 + 0 = 1

Upon hearing that certain numbers did this, a great cry should well up from your soul. And that cry should be, "Okay. And?" Sure, certain numbers will do this square down to one this way. And other numbers will not. Why should it matter at all? Mathematicians are not sure, but they do know that numbers that don't get broken down to one only meet a single fate. Eventually, they get down to four. Once a number hits four, it's locked in an endless cycle. I won't go through the math, but the terms go as follows, 4, 16, 37, 58, 89, 145, 42, 20, 4. And around and around again forever. This is an unhappy number, or some call it a dizzy number.

Before you ask, no use has ever been made of happy numbers. It's just something that mathematicians have unearthed time and time again while digging in their sandpit. But perhaps it should give us pause, in a Lost conspiracy kind of way. A simple process can work every number to one of two conclusions. God's binary code? Or just another weirdness of mathematics?

Top Image: Jin Thai
Via Math World and Articles for Educators.


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