495 (four hundred
ndninety-five) is the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
following
494 and preceding
496. It is a
pentatope number (and so a
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
). The maximal number of pieces that can be obtained by cutting an annulus with 30 cuts.
Kaprekar transformation
The
Kaprekar's routine
In number theory, Kaprekar's routine is an iterative algorithm that, with each iteration, takes a natural number in a given number base, creates two new numbers by sorting the digits of its number by descending and ascending order, and subtracts th ...
algorithm is defined as follows for three-digit numbers:
# Take any three-digit number, other than repdigits such as 111. Leading zeros are allowed.
# Arrange the digits in descending and then in ascending order to get two three-digit numbers, adding leading zeros if necessary.
# Subtract the smaller number from the bigger number.
# Go back to step 2 and repeat.
Repeating this process will always reach 495 in a few steps. Once 495 is reached, the process stops because 954 – 459 = 495.
Example
For example, choose 495:
:495
The only three-digit numbers for which this function does not work are
repdigit
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of repeated and digit.
Example ...
s such as 111, which give the answer
0 after a single iteration. All other three-digit numbers work if leading zeros are used to keep the number of digits at 3:
:211 – 112 = 099
:990 – 099 = 891 (rather than 99 – 99 = 0)
:981 – 189 = 792
:972 – 279 = 693
:963 – 369 = 594
:954 − 459 = 495
The number
6174
6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is renowned for the following rule:
# Take any four-digit number, using at least two different digits (leading zeros are allowed).
# Arrange the digit ...
has the same property for the four-digit numbers, albeit has a much greater percentage of workable numbers.
See also
*
Collatz conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integ ...
— sequence of unarranged-digit numbers always ends with the number 1.
References
*
{{Integers, 4
Integers
fr:Nombres 400 à 499#495
ja:400#481 から 499