- #1

- 60

- 0

I'm working through the book "Problem Solving Strategies" by A.Engel at the moment.

There is an example in the introductory section of the "The Box Principle" chapter, I don't quite understand.

Problem:

A chessmaster has 77 days to prepare for a tournament. He wants to play at least one game per day, but not more than 132 games. Prove that there is a sequence of successive days on which he plays exactly 21 games.

Solution:

Let [itex]a_i[/itex] be the number of games played until the [itex]i[/itex]th day inclusive.

Then [itex]1\le{a_1}<...<a_{77}\le{132}\Rightarrow22\le{a_1+21}<a_2+21<...<a_{77}+21\le{153}[/itex]

Among the 154 numbers [itex]a_1,...,a_{77},a_1+21,...,a_{77}+21[/itex] there are two equal numbers. Hence there are indices i, j, so that [itex]a_i=a_j+21[/itex]. The chessmaster has played exactly 21 games on the days # [itex]j+1,j+2,...,i[/itex].

Well, if I got it right, the boxes are the days and the pearls are the games, spread among the days and whose number per day is not greater than 132.

(Personally, I would expect the converse)

My first question would be: Why is the second sequence formed from the initial added to this, i.e. why is there suddenly a doubled number of games, and hence boxes?