10 people sharing marbles: We have 10 people standing in a row. At first, the total number of marbles the females have were...

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nanase

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We have 10 people standing in a row. At first, the total number of marbles the females have were equal to the total number of marbles the males have. Then each person gave a marble to every person standing to the right of him/her. After that, the total number of marbles females have increased by 25. How many girls are there in the row?

May I have some help how to start approach this kind of algebra work?
I tried let x = number of marbles each man has
let y = number of marbles each woman has
a = number of men
b = number of women
so ax + by = something I don't know
but a + b = 10

how to form an expression for the last sentence ? something something = by + 25
 
Numbers of men and women are not enough to determine the change in the number of marbles: their positions in the row matter too. Here is my first step: number all positions left to right from 0 to 9, then figure out the change in the number of marbles had by the person in the [imath]k[/imath]-th position.
 
from your hint I am thinking that the 1st person will lose one marble.
The 2nd until 9th person will have the same amount ( as they receive one and give away one)
The 10th person will gain one extra marble.

I can't use Arithmetic sequence here right? is setting up algebraic equation possible ?
not knowing male/female in each term is confusing me.
 
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nanaseailie said:
We have 10 people standing in a row. At first, the total number of marbles the females have were equal to the total number of marbles the males have. Then each person gave a marble to every person standing to the right of him/her. After that, the total number of marbles females have increased by 25. How many girls are there in the row?

May I have some help how to start approach this kind of algebra work?
I tried let x = number of marbles each man has
let y = number of marbles each woman has
a = number of men
b = number of women
so ax + by = something I don't know
but a + b = 10

how to form an expression for the last sentence ? something something = by + 25
Click to expand...
To whom does the 10th person give the marble?
 
blamocur said:
I disagree. From your post : "... gave a marble to every person standing to the right of him/her."
Click to expand...
Ah thank you for pointing this out, I misinterpreted thinking they only give to the next person only.
so the 10th will receive 9+8+7+6+5+4+3+2+1 = 45 marbles,
9th will receive 8+7+6+5+4+3+2+1 = 36 marbles.
8th will receive 28 marbles
7th will receive 21
6th will receive 15
5th will receive 10
4th will receive 6
3rd will receive 3
2nd will receive1
1st will receive 0 (as he is the universal donor)
The only information I have that I can not fixate is the total marbles owned by female increased by 25, where can I fit that in?
nor any info how many males/females are there.
 
Most of them don't just receive, but give away as well. How does the total number of marbles change for each person?
 
@blamocur Ahh thank you, I think I did wrong in the adding above as well.

10th will receive 9 marbles and gives out 0 marble. Total net =9
9th will receive 8 marbles, and gives out 1. Total net = 7
8th will receive 7 marbles, gives out 2. Total net = 5
7th will receive 6, gives out 3. Total net = 3
6th will receive 5, gives out 4. Total net = 1
5th will receive 4, gives out 5. Total net = -1
4th will receive 3, gives out 6. Total net = -3
3rd will receive 2, gives out 7. Total net = -5
2nd will receive 1, gives out 8. Total net = -7
1st will receive 0 , gives out 9. Total net = -9
so everything cancels out in the flow of marbles? I am still stuck with the 'puzzle sentence' that the number of marbles female have increased by 25. How do I fit that in?
 
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nanaseailie said:
so everything cancels out in the flow of marbles?
Click to expand...
Which makes sense since all the marbles were re-distributed inside the group.
nanaseailie said:
I am still stuck with the 'puzzle sentence' that the number of marbles female have increased by 25. How do I fit that in?
Click to expand...
Can you find all subgroups of people for which the total increase in the number of marbles is 25? For example, a subgroup of 5th, 7th and 9th person increased their total by -1 + 3 + 7 = 10. You need to find subgroups where this increase is 25.
 
wow wow wow !!!
so adding all the positive exchange (9+7+5+3+1) from the 6th, 7th, 8th, 9th and 10th person gives 25 marbles increase. So that means there are 5 girls? Did I get this right?
 
blamocur said:
You did indeed. And you figured out their positions as a bonus :)
Click to expand...
Thank you so muchhhhh for your guidance and help.
I wonder how you are seem to be able to solve every problem (olympiad types)? I would like to be as good as you.
What is the trick? is it experience? or you have done similar questions? or are you just super good in Maths? or you need high IQ ? or something else?
Many mentors here seem to be able to solve hard questions at one glance.
 
nanaseailie said:
Thank you so muchhhhh for your guidance and help.
I wonder how you are seem to be able to solve every problem (olympiad types)? I would like to be as good as you.
What is the trick? is it experience? or you have done similar questions? or are you just super good in Maths? or you need high IQ ? or something else?
Many mentors here seem to be able to solve hard questions at one glance.
Click to expand...
I don't have an answer. As a kid I enjoyed competitive problem solving, but now I consider this more as a hobby rather than a good educatoin. I'd recommend spending more time and effort on learning interesting new fields (group theory, topology, etc) rather than getting good at solving exotic olympiad-style problems. There is more elegance in those fields than in those artificial problems they feed you in math olympiads. There is nothing wrong with having occasional fun solving interesting puzzles, but I wouldn't spent too much time on them, dispite, or because of, their addictiveness.
 
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