Question

Prove using Pigeon hole principle

a)We have three weeks to prepare for tennis tournament . We will play at least one match each day but no more than 36 matches in total. Show that there is a period of consecutive day where you exactly play 21 matches. b) We now have 11 weeks , We will play at least one match per day up to total of 132 matches. Prove the same conclusion as above

Answer 1

A) let ri be the number of sets played up to and including day i. So

0=r0<r1<r2<...<r21≤360=r0<r1<r2<...<r21≤36

So there are 37 pigeonholes - possible values - for the 22 pigeons - the ri

However, we are looking for pairs i,jwith rj=ri+21 - since such a pair would tell us that 21 sets were played on days i+1 through to j. So if we assume there are no such pairs, we can "combine" the a) pigeonholes i and i+21. So we have the following pigeonholes:

{0 or 21},{1 or 22},...,{15 or 36},{16},...{20}{0 or 21},{1 or 22},...,{15 or 36},{16},...{20}.

And we have 22 pigeons, 21 pigeonholes, giving a contradiction

Sorry i kno only one answer