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.

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