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The restrictions which define an autobiographical number make itstraightforward to find the lowest one.
It cannot be 0, since by definitionthe first digit must indicate the number of zeros in the number. Presumablythen, the smallest possible autobiographical number will contain only one 0.If this is the case, then the first digit must be 1.
10 is not a candidatebecause the second digit must indicate the number of 1s in the number--inthis case, 1. So If the number contains only one zero, it must contain morethan one 1. (If it contained one 1 and one 0, then the first two digits wouldbe 11, which would be contradictory since it actually contains two 1s)
Again,presumably the lowest possible such number will contain the lowest possiblenumber of 1s, so we try a number with one 0 and two 1s. It will be of theform: 12 ... 0 ... Now, there is one 2 in this number, so the first threedigits must be 121. To meet all the conditions discussed above, we can simplytack a 0 onto the end of this to obtain 1210, which is the smallest auto-biographical number.
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