When it comes to peculiar numbers, pi is blasé. Whether you're looking for your next four-digit phone passcode or just something to talk about the next time you bump into a mathematician, we've got you covered with these intriguing collections of digits.
1. The Münchhausen number: 3435
Named after 18th century German nobleman Baron Hieronymus von Münchhausen—whose tall tales were turned into fictional stories that included riding a cannonball and visiting the moon—this number takes its name from its ability to “raise itself.” A Münchhausen number is one that equals the sum of its digits raised to each digit's power. Since 0^0 is not well-defined, the only Münchhausen numbers are 1 and 3435 = 3^3 + 4^4 + 3^3 + 5^5 = 27 + 256 + 27 + 3125.
The individuality of the number alone would be enough to interest us, but since it comes complete with a pun based a fictional character, it’s more than deserving of a place on our list.
2. The Self-Descriptive Number: 6210001000
There is precisely one base-10 number that meets the criteria for a Self-Descriptive number: a 10-digit number where the digits can be numbered 0 to 9, and any digit n is the number of ns in the number:
6 = 0, so there are six zeros in the number.
2 = 1, so there is one two in the number.
1 = 2, so there are two ones in the number.
0 = 3, so there are zero threes in the number.
0 = 4, so there are zero fours in the number.
And so on.
3. Kaprekar’s Constant: 6174
Why constant? Because if you perform the following process, Kaprekar's routine, the operation will always yield 6174 in no more than seven iterations.
Take any four-digit number, using at least two different digits. (Leading zeros are allowed, to keep the number of digits at four.)
Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.
Subtract the smaller number from the bigger number.
Go back to step 2.
To illustrate the process, we’ll perform it on 7455.
7554 - 4557 = 2997
2997 → 9972 - 2799 = 7173
7173 → 7731 - 1377 = 6354
6354 → 6543 - 3546 = 3087
3087 → 8730 - 0378 = 8352
8352 → 8532 - 2358 = 6174, in six iterations of the routine.
Go on. Try it yourself. We'll wait.
4. The Hardy–Ramanujan number: 1729
This number got its start when British mathematician G. H. Hardy visited Indian mathematician Srinivasa Ramanujan while Ramanujan was ill in the hospital. Hardy later recalled:
I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. ‘No,’ [Ramanujan] replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.’
The two cubes are 1729 = 1^3 + 12^3 = 9^3 + 10^3. Numbers that are the smallest number expressible as the sum of two positive cubes in n distinct ways are called “taxicab numbers” for just this reason.
5. The Golden Ratio: 1.618...
This one is more of a mathematical expression where, if variables a and b are greater than zero, a + b : a :: a : b, which basically means that the sum of the variables is to the first variable as the first variable is to the second.
This ratio has been known since the Ancient Greeks, and some people see it represented in architectural works. For instance, the Great Mosque of Kairouan is often cited as a nearly perfect representation (but the jury is still out). Somewhere along the line people got the idea that this ratio is super sexy—though Euclid himself recorded it simply as “an extreme and mean ratio” in the Elements—and some modern designers deliberately incorporate it into their designs. More recently, its been found in certain patterns in nature.