DLS
  • DLS
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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DLS
  • DLS
@ganeshie8 @ikram002p
DLS
  • DLS
I have an idea about this but I'm not sure how to put it in words.
ganeshie8
  • ganeshie8
Nice, before that, could you explain me your interpretation of the question ?

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More answers

DLS
  • DLS
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
  • DLS
@ganeshie8 is it correct ? :/
ganeshie8
  • ganeshie8
that looks good to me !
DLS
  • DLS
so..next :o
ganeshie8
  • ganeshie8
we need to choose the pigeons and holes a bit carefully
ganeshie8
  • ganeshie8
lets not rush deciding on that yet
DLS
  • DLS
alright..
ganeshie8
  • ganeshie8
but do keep that in mind, we will need to choose them once we're ready
DLS
  • DLS
sure :D
ganeshie8
  • ganeshie8
familiar with partial sums notation right ?
DLS
  • DLS
maybe I don't know it by name, show me ?
ganeshie8
  • ganeshie8
\(S_n\) represents the \(n\)th partial sum, the total number of games played over the first "n" days
DLS
  • DLS
alright.
ganeshie8
  • ganeshie8
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
  • DLS
yep
ganeshie8
  • ganeshie8
\(S_n\) is total games played during first \(n\) days \(S_{m-1}\) is total games played during first \(m-1\) days
ganeshie8
  • ganeshie8
subtracting gives you the total number of games played from "m"th day to "n"th day
DLS
  • DLS
yeah..where n>m
ganeshie8
  • ganeshie8
yes
ganeshie8
  • ganeshie8
we want to show that \(S_n-S_{m-1}\) is \(14\) for some integers \(n\) and \(m\)
DLS
  • DLS
yes yes
ganeshie8
  • ganeshie8
lets try and figure out how to approach this
ikram002p
  • ikram002p
45 game per month or day ?
ikram002p
  • ikram002p
i think it should be month though :O
DLS
  • DLS
45 overall..maximum games..per month
ikram002p
  • ikram002p
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. |dw:1449430580324:dw|
ikram002p
  • ikram002p
i'm I making any sense xD
ikram002p
  • ikram002p
this is the minimum case which any 14 consecutive days they played exactly 14 games. |dw:1449430993787:dw|
DLS
  • DLS
yep..makes sense
DLS
  • DLS
but how do we prove it in words :|

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