The number I am thinking of has to contain at least five digits, as for any smaller number the total number of digits I write down cannot be exactly four times my number, as many of the numbers only contribute one to three digits each.

The total number of digits I write down up to 9,999 = 9 x 1 + 90 x 2 + 900 x 3 + 9,000 x 4 = 38,889

The total number of digits up to my number n, is therefore 38,889 + 5(n – 9,999) which must equal 4n.

Rearranging this equation gives 5n – 11,106 = 4n and so n = 11,106.