Physicist: If you’ve ever sat next to someone as they rang an almost-empty ice cream bowl like church bells on Sunday, then you’ve probably asked yourself some variation of this question.
Clearly, the issue is that a spoon and a bowl aren’t osculating curves. The radius of a spoon’s curvature is typically less than an inch while a bowl’s is several inches. With different curvatures, when you drag a spoon across the bottom of an ice-cream-laden bowl, they only contact each other at a tiny point and you can only remove ice cream along a thin strip (Matching the curvatures is why you instinctively start using the spoon sideways when the ice cream is low). So if you wanna get your bowl perfectly clean, you gotta figure out how wide that strip is.
Around the point of contact there’s a gap between the spoon and bowl that gets bigger the farther away you get. But so long as the ice cream molecules are bigger than that gap, they’ll be caught. Water, at about a quarter of a nanometer across, is the smallest molecule in ice cream’s molecular menagerie.
The bottom of a circle with a radius R can be closely approximated by a parabola of the form (There’s nothing too special about parabolas; every curve that doesn’t have sharp corners can be closely approximated by circles and vice versa, it’s just that parabola math is easy). Spoons have a radius of around 1 cm. Bowls have a radius of around 7 cm. The distance from the center (the point of contact) to the edge of the strip, x, such that water molecules won’t be able to slip through the gap between the bowl and the passing spoon is given by . Solve for x and you get , so the strip should be about 5 micrometers across; on the order of a tenth of a hair’s width.
The wide ice-cream-free swaths you see in practice aren’t regions where all of the ice cream has been removed, but regions where it’s been left thin enough to see though. So, if you really wanted to scrape a bowl clean of every molecule of ice cream, you’d need to carefully “scan” one strip after another, about four million and change times. Just to make sure you don’t miss any (else you and your IC OCD would be forced to start over), it couldn’t hurt for those strips to overlap. Call it an even ten million passes. At, say, two passes per second (the approximate speed of a kid charging headlong into a brain freeze), it would take you almost two months to scour your bowl clean. About a week into that, you’ll get the overwhelming urge to ask for seconds. Or at least do something else.
So far, this is all a bit idealized. The Platonic perfection of parabolas doesn’t apply to actual spoons and bowls. If you zoom in close enough, ceramics are like moonscapes with plenty of room for ice cream to hide and the tip of a spoon looks more like a stainless steel mountain range. Truly, a rocky road.
As undeniably clever as the approach above is, that “10,000,000 passes with a spoon” estimate is built on unrealistic premises. Sadly, you can’t eat all the ice cream in a conventional bowl using compulsive spooning alone.
But that doesn’t mean it’s impossible eat all of the ice cream, you just have to stretch the rules a little. For example, you could eat as much as you would like, then put the bowl into a kiln. By and large, organic molecules (sugar, fat, cellulose, etc.) burn at temperatures below 500°F and turn into new compounds, and a good kiln heats things up to 1700°F. A ceramic bowl would survive the heat intact, but the chemicals that make ice cream ice cream (as opposed to ash) wouldn’t.
So if you define “eating all the ice cream in the bowl” as “there was ice cream, I ate ice cream, now it’s all gone”, a kiln is a good way to do it. But if you think destroying the evidence is somehow cheating, there is one more, perhaps not as delicious, alternative guaranteed to work.
Eat the bowl.
The empty ice cream bowl picture is from here.
The extreme close-up of a ceramic is from here.
The kiln picture is from here.
The Willy Wonka picture is from Willy Freaking Wonka.