During a month with 30 days a baseball team plays at least 1 game a day, but no more than 45 games. Show that there must be a period of some number of consecutive days during which the team must play exactly 14 games.

Question

dls · Answer

@ganeshie8 @ikram002p

dls · Answer

I have an idea about this but I'm not sure how to put it in words.

ganeshie8 · Answer

Nice, before that, could you explain me your interpretation of the question ?

dls · Answer

Let the games played till ith day be x1+x2+x3..+xi..(i=1 to 30)
Where x1,x2,...x30>=1
and what its asking to prove is that there exists two indices i and j such that summation from i=1 to j (xi) = 14.
I guess we can keep days as pigeons.

dls · Answer

@ganeshie8 is it correct ? :/

ganeshie8 · Answer

that looks good to me !

dls · Answer

so..next :o

ganeshie8 · Answer

we need to choose the pigeons and holes a bit carefully

ganeshie8 · Answer

lets not rush deciding on that yet

dls · Answer

alright..

ganeshie8 · Answer

but do keep that in mind, we will need to choose them once we're ready

dls · Answer

sure :D

ganeshie8 · Answer

familiar with partial sums notation right ?

dls · Answer

maybe I don't know it by name, show me ?

ganeshie8 · Answer

$S_n$ represents the $n$th partial sum, the total number of games played over the first "n" days

dls · Answer

alright.

ganeshie8 · Answer

using that notation, the total number of games played from "m"th day to "n"th day can be represented as : 
$$S_n-S_{m-1}$$

dls · Answer

yep

ganeshie8 · Answer

$S_n$ is total games played during first $n$ days
$S_{m-1}$ is total games played during first $m-1$ days

ganeshie8 · Answer

subtracting gives you the total number of games played from "m"th day to "n"th day

dls · Answer

yeah..where n>m

ganeshie8 · Answer

yes

ganeshie8 · Answer

we want to show that $S_n-S_{m-1}$ is $14$ for some integers $n$ and $m$

dls · Answer

yes yes

ganeshie8 · Answer

lets try and figure out how to approach this

ikram002p · Answer

45 game per month or day ?

ikram002p · Answer

i think it should be month though :O

dls · Answer

45 overall..maximum games..per month

ikram002p · Answer

well we have at least 30 games per month, assume each day they played 2 games then we would got 60 games per month >45, which means 2 is the maximum games played per day.
now assume they played the max of 45 games means we are gonna arrange at most 45 games of 1 and 2 over 30 days. 
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