Computational recreations: the n days of Christmas

Abstract

In what will likely become a regular feature, this issue of Crossroads introduces the Computational Recreations column. These columns will include puzzles, games, and brainteasers intended to challenge and tickle the mind. In the well-known song The Twelve Days of Christmas, each verse builds upon the previous one to describe an ever-increasing influx of gifts. But why stop after a mere twelve days when we can extend the song? Instead of a fixed number, we can continue for some arbitrary number of days! This new song, The n Days of Christmas, continues the pattern established in the original version: 1. On the first day, you receive a present of type T 1. 2. On the second day, you receive two presents of type T 2 , plus one of type T 1. 3. On the third day, you receive three presents of type T 3 , two of type T 2 , and one of type T 1. ... 4. On the n th day, you receive n presents of type T n , plus n-1 of type T n-1 , n-2 of type T n-2 , and so forth, down to one present of type T 1. Questions 1. How many presents do you receive on day n?