Values of n?

Question

For each positive integer n, let S(n) denote the sum of the digits of n. For how many values of n is n+S(N)+S(S(n)) = 2007?

Answer 1

Three


n + S(n) + S(S(n)) = sum
0 + 0 + 0 = 0
1+ 1 + 1 = 3
2 + 2 + 2 = 6
...
10 + 1 + 1 = 12
19 + 10 + 1 = 30
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above was to show how the sum works


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2001 + 3 + 3 = 2007
2002 + 4 + 4 = 2010
...
2007 + 9 + 9 = 2025

n cannot be bigger than 2001

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1969 + 25 + 7 = 2001
1963 + 19 + 10 = 1992
n must be bigger than 1970
(At least we know that the number of n's is finite and much less than 30)

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1979 + 26 + 8 = 2013
1978 + 25 + 7 = 2010
1977 + 24 + 6 = 2007

1986 + 24 + 6 = 2016
1985 + 23 + 5 = 2013
1984 + 22 + 4 = 2010
1983 + 21 + 3 = 2007

1990 + 19 + 10 = 2019
1991 + 20 + 2 = 2013

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