Question

A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?

Answer

Solution:

Write $1$ for a house that gets mail and $0$ for one that does not. Valid patterns are binary strings of length $19$ with no two consecutive $1$ s and no three consecutive $0$ s. Let $A_n,$ $B_n,$ $C_n$ count valid length- $n$ strings ending in $1,$ in exactly one $0,$ and in exactly two $0$ s. A $1$ may follow either kind of $0$ -ending, a single $0$ may follow a $1,$ and a second $0$ may follow a single $0:$ $A_n = B_{n-1} + C_{n-1}, \qquad B_n = A_{n-1}, \qquad C_n = B_{n-1}.$

Starting from $A_1 = B_1 = 1,$ $C_1 = 0$ and iterating, the totals $A_n + B_n + C_n$ run $2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351.$

For $n = 19$ the count is $351.$

2001 AIME I Problem 14

Solution:

Problem 14 in Other Years