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athenaisis84

I HAvE TWENTY OF THESE DUE BY TUESDAY HELP:

 

 

Mr. Whitlock is playing a game with his math class to teach them about money. Mr. Whitlock’s
math class consists of n ≥ 2 students, whom he has numbered from 1 to n. Mr. Whitlock gives mi ≥
0 dollars to student i, for each 1 ≤ i ≤ n, where each mi is an integer and m1 + m2 + ··· + mn ≥ 1.
We say a student is a giver if no other student has more money than they do and we say a student
is a receiver if no other student has less money than they do. To play the game, each student who
is a giver, gives one dollar to each student who is a receiver (it is possible for a student to have a
negative amount of money after doing so). This process is repeated until either all students have
the same amount of money, or the students reach a distribution of money that they had previously
reached.
(a) Give values of n, m1, m2,...,mn for which the game ends with at least one student having a
negative amount of money, and show that the game does indeed end this way.
(b) Suppose there are n students. Determine the smallest possible value kn such that if m1 +m2 +
··· + mn ≥ kn then no player will ever have a negative amount of money.
(c) Suppose n = 5. Determine all quintuples (m1, m2, m3, m4, m5), with m1 ≤ m2 ≤ m3 ≤ m4 ≤
m5, for which the game ends with all students having the same amount of money

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catdragon13873

I'm not doing your homework for you. Nice try though.

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JADE SQUIDSHIPQUAKE

oceans rise, empires fall

we have seen each other through it all

 

for even if i‘m far away i hold you in my heart

 

no matter where you go

you’ll never be alone

 

toffee’s still a walnut

 

I'm going down with this ship

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athenaisis84

internally cries

 

 

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lllY'all are cutelll

Fluffeh Lotus

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) (

( )

#//#

Up in the sky,   they twinkle bright,

The stars lead the way, into the night,

Giving way to the lurking shadows of night.

 

*facepalmqueen... or am  I?

 

 

 Join night clan!

 

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tricksterangel48

I could probably make sense of it if I tried for a while, but I’m not going to. It’s Saturday. I do not do maths on Saturday.

What grade are you even in, if you don’t mind my asking?

And why would anyone possibly need to know that?

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ceol

infp // libra // hufflepuff // bisexual

ill see you in the future when were older

  and we are full of stories to be told

() ~remember chester bennington~

gang: dan, danny, grey [formerly dan II], aleks

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tricksterangel48

wAIT IGNORE MY OTHER POST I FOUND THE SOLUTION ONLINE
HAVE FUN MAKING SENSE OF THIS:

 

(a) Suppose there are 5 students and the starting amounts are: (0, 0, 0, 1, 2). As the game progresses, the amounts of money become: (1,1,1,1,−1) then (0,0,0,0,3) and finally (1, 1, 1, 1, −1) at which point the game ends.

 

(b) Consider a game with n players that starts with the initial distribution of one player having n − 2 dollars and all the rest having n − 3. After the first turn, the first player will have −1 dollars, thus n − 2 + (n − 3)(n − 1) = n2 − 3n + 1 is not sufficient.


We observe that the only way for a person to end up with negative money is if they were the giver and gave away more money than they had. Suppose instead that the sum of all the money is n2 − 3n + 2. By the pigeonhole principle, at every stage of the game, we must either have at least one student with at least n − 1 dollars or at least two students with exactly n − 2 dollars. In the first case, since the giver is giving one dollar to at most n − 1 people, they will not end the turn with a negative amount of money. Similarly, if there are at least two students with n − 2 dollars, then they will each give one dollar to at most n − 2 students and will not end the turn with a negative amount of money. Thus kn = n2 − 3n + 2.

 

(c) We observe that it is clearly necessary for m1 +m2 +m3 +m4 +m5 to be a multiple of 5 in order for the students to all have the same amount of money, so we will assume that this is the case. Let m be the average amount of money. We see that if a student ever has exactly m dollars, they will have m dollars for the remainder of the game, since either everyone will have m dollars or there will be someone with more money and someone with less money.


We first consider the following scenarios:

 

• m1 =m2 =m3 =m4

 

• m1

 

• m1 =m2 =m3

 

• m1 =m2 < n3 < n4 = n5 dollars respectively, with n3 = n2 + 1. Since the total amount of money is divisible by 5 we get that n4 = n5 = n2 +2p+2 for some non- negative integer p. If p > 0 then the dollar amounts after the next turn is (n2 +2, n2 + 2, n2+1, n2+2p, n2+2p) and then (n2+2, n2+2, n2+3, n2+2(p−1)+4, n2+2(p−1)+4) which is the same as (q2, q2, q2 +1, q2 +2(p−1)+2, q2 +2(p−1)+2). Thus, eventually we will get to a point where the players have (m−1,m−1,m,m+1,m+1) dollars respectively, and this game will not end with all players having the same amount of money.

 

• m1 =m2 < k, in which case we will be in the first scenario and the game will end with all players having the same amount of money.

 

We now seek to reduce all cases to one of the above scenarios. Let us assume WOLOG that m2 −m1 ≤ m5 −m4. This means that on the first m2 −m1 turns, player 5 will have given m2 − m1 dollars to player 1. So after this point the respective dollar amount for the players will be m2, m2, m3, m4, m′5, where m′5 = m5 − m2 + m1.

 

If m′5 − m4 ≥ 2(m3 − m2) then after the next m3 = m2 turns, the first three players will have the same amount of money so the game ends with all players having the same amount of money.

 

If k = m′5 − m4 is even then after the next k/2 moves players 4 and 5 will both have m4 dollars and the first two players will have (m1 + m2 + m5 − m4)/2 dollars. This will end with all players having the same amount of money if and only if min(m4 − m3 , m3 − (m1 +m2 +m5 −m4)/2 is even. Note that k is even if and only if m1 +m2 +m4 +m5 is even.

 

If k is odd, then player 5 will give money to players 1 and 2 until they have only 1 dollar less than player 4. Then player 4 and 5 will take turns giving money to players 1 and 2 until players 1 and 2 have the same amount of money as player 3, or one of players 4 and 5 has the same amount of money as player 3. If player 1 and 2 have the same amount as player 3, the game will end with all players having the same amount of money. This occurs when m5 +m4 −2m3 > 2m3 −m2 −m1. If either 4 or 5 has the same amount has 3, the game will not end with all the players having the same amount of money. Thus, we see the game will end with all players having the same amount of money when m2 − m1 ≤ m5 − m4 and one of the following 3 conditions hold


• m5 − m4 ≥ 2m3 − m2 − m1
• m1 +m2 +m4 +m5 and min(m4 −m3,m3 −(m1 +m2 +m5 −m4)/2) are both even
• m1+m2+m4+m5 is odd and m5+m4−2m3 >2m3−m2−m1
or when m2 − m1 > m5 − m4 and one of the following 3 conditions hold
• m2 − m1 ≥ m5 + m4 − 2m3
• m1 +m2 +m4 +m5 and min(m3 −m2,(m1 −m2 +m4 +m5)/2−m3) are both even

• m1+m2+m4+m5 is odd and 2m3−m2−m1 >m5+m4−2m1.

 

^^^^^^^

 

I have no idea what this is even, but I hope this helps. I wish you luck on whatever the heck you’re doing in maths.

I also dearly hope we never have to do this.
Anyway just copy-and-pasting this gave me the worst headache so I’m just gonna go

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ceol

infp // libra // hufflepuff // bisexual

ill see you in the future when were older

  and we are full of stories to be told

() ~remember chester bennington~

gang: dan, danny, grey [formerly dan II], aleks

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amberpegasus203

I'm sorry, but I can't help you on this. (I'm not that good at math to begin with...)

But I can say that this Mr. Whitlock should probably know that there are better ways to teach students about money.

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Raven Tsakaryn

   ~est. April 2014~

 

Oh, how I've missed you, my friend...

Time has changed us both.

But deep down, our hearts remain the same.

So, no matter where life takes us, remember...

 

 

 

 

 

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butterflydusk36

i have no idea but you should probably listen to what ceol said because she seems to be smart and willing to do things unlike me hahahaha funny joke becoming depressing joke

anyways listen to what raven said too, those are wise words 

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 she grew a lust to bury all underneath

 she cut to their heart,

 she bled them dry

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livelyfire22

I love math! And this seems to be around where I'm at, but I'm not gonna help ya cheat, sorry.

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catdragon13873

Agreed.

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JADE SQUIDSHIPQUAKE

oceans rise, empires fall

we have seen each other through it all

 

for even if i‘m far away i hold you in my heart

 

no matter where you go

you’ll never be alone

 

toffee’s still a walnut

 

I'm going down with this ship

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tricksterangel48

Ever tells the best jokes. “Ceol seems smart and willing to do things”

Honestly it’s like saying “Donald Trump seems intelligent and rational, he’ll be a great president”

HA

no

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ceol

infp // libra // hufflepuff // bisexual

ill see you in the future when were older

  and we are full of stories to be told

() ~remember chester bennington~

gang: dan, danny, grey [formerly dan II], aleks

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butterflydusk36

let's all tell really depressing jokes at a really bad time and all laugh 

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Ever After~

 i knew a lady lost the forest for the trees

 she grew a lust to bury all underneath

 she cut to their heart,

 she bled them dry

 she kept her fire burning up to the sky

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airportwinter35

What grade are you in? 

I had solved questions like that one hours ago.

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 t Jacob (major and mostly used) t

 

      

   Anything interesting

 

Stay Calm, Study Yourself, Stop Wasting Time.

These are all 'St process of life. 

                                                                                           

 

 

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airportwinter35

Oh, it's Sunday night in West Pacific Timezone!

Ask to residents below:

Australia

Korea

Japan

China

 

If you want to get a answer, ask some genius kids in world. Plz mind their time.

 

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 t Jacob (major and mostly used) t

 

      

   Anything interesting

 

Stay Calm, Study Yourself, Stop Wasting Time.

These are all 'St process of life. 

                                                                                           

 

 

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athenaisis84

 

Nvm guys I solved it

 

 

it's just math contest prep tha tmakes me hate my life

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lllY'all are cutelll

Fluffeh Lotus

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( )

#//#

Up in the sky,   they twinkle bright,

The stars lead the way, into the night,

Giving way to the lurking shadows of night.

 

*facepalmqueen... or am  I?

 

 

 Join night clan!