the "3" sequence and the length of the "6" sequence are the same) are fixed points of the Kaprekar mapping. b = 2k It can be shown that all natural numbers are fixed points of the Kaprekar mapping in even base for all natural numbers . Kaprekar's constants and cycles of the Kaprekar mapping for specific base b All numbers are expressed in base , using A−Z to represent digit values 10 to 35. Kaprekar's constants in base 10 Numbers of length four digits In 1949 D. R. Kaprekar discovered that if the above process is applied to base 10 numbers of four digits, the resulting sequence will almost always converge to the value 6174 in at most eight iterations, except for a small set of initial numbers which converge instead to 0. The number 6174 is the first Kaprekar's constant to be discovered, and thus is sometimes known as Kaprekar's constant. The set of numbers that converge to zero depends on whether leading zeros are discarded (the usual formulation) or are retained (as in Kaprekar's original formulation). In the usual formulation, there are 77 four-digit numbers that converge to zero, for example 2111. However, in Kaprekar's original formulation the leading zeros are retained, and only repdigits such as 1111 or 2222 map to zero. This contrast is illustrated below: Below is a flowchart. Leading zeros are retained, however the only difference when leading zeros are discarded is that instead of 0999 connecting to 8991, we get 999 connecting to 0. Numbers of length three digits If the Kaprekar routine is applied to numbers of three digits in base 10, the resulting sequence will almost always converge to the value 495 in at most six iterations, except for a small set of initial numbers which converge instead to 0. The set of numbers that converge to zero depends on whether leading zeros are discarded (the usual formulation) or are retained (as in Kaprekar's original formulation). In the usual formulation, there are 60 three-digit numbers that converge to zero, for example 211. However, in Kaprekar's original formulation the leading zeros are retained, and only repdigits such as 111 or 222 map to zero. Below is a flowchart. Leading zeros are retained, however the only difference when leading zeros are discarded is that instead of 099 connecting to 891, we get 99 connecting to 0. Other digit lengths For digit lengths other