Odd odometer reading matched at checkout

Published: Thursday, Dec. 20, 2007 12:13 a.m. MST
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Q: From a Utah reader: "My husband and I were driving when he happened to spot the odometer reading 22,222. Then we drove immediately to a store where we made multiple purchases and, yup, at the checkout the total was $222.22.

We are wondering: Is there any way to figure the odds of the numbers coming up like this?"

A: There are lots of possible assumptions and answers, suggests University of California-Los Angeles mathematician James Ralston.

For example, ignoring the decimal and assuming all five-digit numbers between 10000 ($100.00) and 99999 ($999.99) are equally likely, the chance is 1 in 90,000, since there are 90,000 such numbers.

If you broaden this to include the UNLIKELIHOOD of spotting the 22,222 on the odometer, things get even more interesting. Let's call these "monodigital" readings, says College of Charleston mathematician Alex Kasman. The probability of spotting one of these at any given time is small. Starting out, a car runs through 11 miles, 22, 33 ... 111, 222, 333 ... and so on. Drive 100,000 miles and you'll turn over 36 monodigitals. "They're certain to occur, though you probably won't notice most of them."

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Now suppose you make a game of watching for these, and every time you spot one you pull over to make some random purchases. Can you reproduce the checkout feat of the Utah reader? Not likely, but not impossible either, says Kasman.

If you spend $10 to $1000, there are four monodigitals — $22, $22.22, $222 and $222.22 out of 99,001 possible prices — to match 22,222 miles. Thus the probability is 4/99,001, or just a bit more than a one-in-25,000 chance.

Probably the game will drive you crazy first ....

Q: Why is ice so slippery? Better watch your step on this one ... (a) pressure melting puts water underfoot, (b) frictional heating under skis or skates creates the glide, (c) it's just ice's icy nature

A: (a) and (b) used to be given, but it's now known that the pressure under a walker's boots or the friction of skate blades against ice doesn't play much of a role in melting, reports Physics Today.

As early as the 1850s Michael Faraday tested ice cubes that were sticking together and concluded that ice surfaces must just inherently consist of a thin film of water, at whatever temperature. So mark c). Even ice at minus 200 F has a "quasi-fluid layer" that makes the stuff slippery, says Exploratorium.edu. This may help explain the "fast ice" and "slow ice" of hockey: As the ice gets warmed, the number of slippery layers increases, until skaters need to "slosh" through so many layers that the friction slows them down.

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