Eli5: Why are there only 88 narcissistic numbers in base 10?
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If you count 0, there are 89, 88 if you don’t count 0.
This isn’t quite ELI5, but nor is the concept of narcissistic numbers, so I won’t tell if you don’t..
Suppose you have an n-digit number, and it’s narcissistic. Since it’s an n-digit number, it’s at least 10^(n-1). However, the sum of its digits is at most nx9^(n). For n bigger than 60, it turns out that n x 9^(n) is always less than 10^(n-1). So there can’t be any narcissistic numbers with 60 or more digits.
For fewer digits, well, very loosely, if you pick a random number, the “chance” of the sum coming out exactly right is tiny – about 1 in 10^(n). But you get about 10^(n) tries, since there’s roughly that many n digit numbers, so you’ll get about 1 narcissistic number with n digits. These are *extremely* loose and hand-wavy calculations, but all they show is that 88 is roughly reasonable.
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