Can you find a seven digit number which describes itself. The first digit is the number of zeros in the number. The second digit is the number of ones in the number, etc.
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Start with the theory that we'll have 6 zeroes - that's 6000000.
But now you need a one in the column for the number of sixes - that's 5000001.
We now don't have a six but we have a 5 and a 1 - that's 5100010.
Now we only have four 0s - 4100010.
Now no 5s but a 4 - 4100100.
Now two 1s but not accounted for. This self corrupts. Follow it through slowly! If we replace the 1 in the 1 column with a two, we'll need to account for the 2. That'll use a digit so we'll reduce the 0 count to 3. We now have a 3.
3211000.
Three zeroes. Two ones. One two. One three. No fours, fives or sixes!
Good luck getting your head around this!
I must say, I absolutely love this question!!