# Self-descriptive Strings

This is a slight modification of the Self-documenting sentences applet. This one is described in M. Gardner's latest book. It appears that the two developments have proceeded in parallel. Gardner presents a short version. (See also [Roberts, p. 84]).

N digits in base N are written in a row and indexed by the numbers 0, 1, ..., N-1. The digit #i is interpreted as stating the total number of digits in the row equal to i. The implied assertion may or may not be correct. Find a self-descriptive sequence of digits that assert its own numbers.

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According to Gardner, there have been several publications tackling this problem. I'll have to look them up in a library. Meanwhile, I'll state the declared results.

1. There is only one solution in base 10: 6,210,001,000

2. In bases 4 through 9 the only answers are 1210, 2020, 21200, 3211000, 42101000, 521001000.

3. For all bases above 6 there is one solution, of the form R21(0...0)1000, where R is four less than the base, and the number of zeros inside the parentheses is seven less than the base.

### References

1. M. Gardner, The Colossal Book of Short Puzzles and Problems, W. W. Norton & Company, 2006, p. 8
2. J. Roberts, Lure of the Integers, MAA, 1992