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## Main Question or Discussion Point

Hi!

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?

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?