• By
  • Carl Bialik

I have a phone number that is right down the center of the phone. Example: 255-8025. I was just wondering how many different seven-digit combinations there are that are right down the center? I am in a fight with someone over it.

– A reader

Lest punches be thrown, I’m answering this question: 12,032, excluding numbers that start with 555, because these generally are reserved for information services and aren’t available for residential lines.

Here’s how I arrived at that number: There are four digits down the center of the typical phone keypad, 0, 2, 5 and 8. But the first of the first three digits of a phone number, known as the prefix, can’t be a 0 or 1, according to John Manning, director of the North American Numbering Plan Administration, who walked me through the numbering system. So that leaves three possibilities for that digit, and four for the next two digits, yielding 3*4*4, or 48, possible prefixes down the center. Exclude 555, and you get 47 prefixes. The next four digits, known as the station number, can each be any of the four center digits, so there are 4^4 or 256 possibilities. And 47*256 is 12,032 — or roughly 0.15% of the roughly 7.86 million phone numbers available in a given area code.

That squares with a simpler method of approximating the probability of getting such a number by chance: Figure that for each of the seven digits, there’s a 40% chance it’s down the center, and raise 0.4 to the seventh power. That yields 0.16%.

So such phone numbers are unusual, but not impossible — slightly more likely than flipping a coin 10 times and getting tails each time.

The question asked only about the seven digits of a phone number. How about the area code? That, like the prefix, can’t start with a 0 or 1. So that leaves 800 possible area codes. But 72 of these are what’s known as “easily recognizable codes,” because they end with identical double digits other than 11; 456 is also reserved, though not all sets of consecutive digits are (234 is in northeast Ohio). These aren’t assigned to geographical areas but are reserved for other purposes. (Just eight have been put to use such far, such as 800 and 877; others, including 722, are idle.) The eight three-digit numbers starting with a digit between 2 and 9, inclusive, and ending in 11 are also unavailable because they serve as three-digit dialing codes rather than prefixes.

We’re left, then, with 719 possible area codes. Three digits down the center of the keypad can start the prefix — 2, 5 and 8 — four can follow it, and three can end it (the digit selected for the middle of the prefix isn’t available, since the last two can’t be identical). That leaves 3*4*3, or 36, possible area codes down the center of the phone — or 5% of the total available. Just 330 geographical area codes have been assigned, of which 4.8% are down the middle. Not each area code is equally likely — some, like 212, fill up and even need to be supplemented with a new area code. But if we assume, for simplicity, that such differences even themselves out, and that numbers are randomly assigned, then the probability of a 10-digit phone number with all digits in the center of the keypad is 0.007%, or slightly more likely than flipping a coin 14 times and getting tails each time.

Thanks to John Cropper, founder of the informational Web site AreaCode-Info.com