48th Putnam 1987 Problem A2 An infinite sequence of decimal digits is obtained by writing the positive integers in order: 123456789101112131415161718192021 ... . Define f(n) = m if the 10 n th digit forms part of an m-digit number. For example, f(1) = 2, because the 10th digit is part of 10, and f(2) = 2, because the 100th digit is part of 55. Find f(1987). Solution
: 1984. Easy. The first 9 numbers have 1 digit, the next 90 2 digits, the next 900 3 digits and so on. So the m-digit numbers and below constitute the first 9(1 + 2.10 + ... + m.10 m-1 ) = 9(1 - (m+1) 10 m + m 10 m+1 )/(1 - 10) 2 = 1/9 (1 - (m+1) 10 m + m 10 m+1 ) digits. For m = 1983 this is about 0.2 10 1987 and for m = 1984 about 2 10 1987 . So the 10 1987 th digit is in a 1984-digit number. 48th Putnam 1987 © John Scholes jscholes@kalva.demon.co.uk 7 Oct 1999