Far from being a white Christmas around here, things are shaping up to be a sort of greyish-green muddy Christmas. I’m disappointed, my son is disappointed, and my wife is thrilled – she doesn’t like snow. But we did get a few flurries last week. At that time, my son stood at the sliding glass door of our condo and stared out. “It’s snowing, daddy!”

“I know,” I say, looking up from my book.

“But it’s not enough to make a snowman,” he adds, sounding a little disappointed.

Glancing outside, I see that there isn’t even enough snow to *dust* the ground. It’s melting before it lands, in some cases. “No,” I agree, “there isn’t.”

“It’ll take six million!” he declares. “Seven, eight million!”

At this moment, I’m pretty sure I see where he’s going with this. But I ask anyway. “Eight million what?”

“It’ll take eight *million* snowflakes to make a snowman!”

Well, that’s not *precisely* a question. But I’ll run with it.

It turns out that there are a few ways to figure the answer. For laughs, I tried punching the question “how many snowflakes in a snowman” into Wolfram Alpha. Believe it or not, I got an answer: an average of 58 million (5.8 x 10^{7}), with a range of 54 million to 61 million (5.4 x 10^{7} to 6.1 x 10^{7}). The calculations assumed that a “typical snowman” could be modeled by a cylinder with a volume of 40 cubic feet, and that a snowflake could be modeled as a sphere with a volume of 0.0000004 cubic feet (4.0 x 10^{-7}) and a packing density of 0.56 to 0.64.

Another way to do this is to figure out the mass of a snowflake. Amusingly enough, The Physics Factbook actually has an entry titled Mass of a Snowflake. It tells us the following:

- The mass of a water molecule is 2.992 x 10
^{-26}kg. - A typical snow crystal may contain 10
^{18}water molecules. - A typical snowflake is made of 100 snow crystals

The end result is that a typical snowflake weighs about 3 mg. There’s 28.3495 grams in an ounce, which is the same as 28,349.5 mg, so that gives us 9,449.8 snowflakes per ounce. There’s 16 ounces in a pound, so that gives us 151,197 snowflakes in a pound of snow.

Here’s where it gets tricky. As far as I can tell, there is no such beast as a “regulation size snowman”. So I’m going to do a lot of estimates here. Let’s assume a three-tier snowman, with the middle tier about two-thirds the size of the base, and the top tier about half the size of the base. And let’s assume my son is building it. My son weighs 49 pounds, and 15-20 pounds seems to be about the limit of what he can lift without help. So, let’s assume he starts by making a 18 pound base (near the upper end of what he seems to be able to handle, because rolling is easier than lifting). That gives him a 12 pound middle and a 9 pound top. So, 39 pounds of snow in all. 39 times 151,197 gives us 5,896,696 snowflakes. Less than the Wolfram Alpha estimate, yes, but I don’t think my son’s snowman is 40 cubic feet of snow, either.

Interestingly enough, he’s not far off from his own estimate of six to eight million snowflakes. Eight million snowflakes would only add another 13.9 pounds (2,103,304 snowflakes/151,197 snowflakes) to the snowman. That would give us a snowman with a 24.4 pound base, a 16.3 pound midsection, and a 12.2 pound head. It might be slightly tough going for my son, but I suspect he could still build that.

For laughs, the hypothetical 39 pound snowman is composed of 589,668,300,000,000,000,000,000,000 water molecules. Numbers like that are the clearest explanation of why exponents were invented.