Sharing Turnips 2
There a 2 rival Turnip growing families, living in Green and Blue houses respectively. Everyone in each house has their own Turnip patch, growing Golden Turnips ( ) and regular Turnips ( ). Using this currency, for some unknown integer .
It is just coming up on harvest season, and both families have run out Turnips in their storage. They decide to get greedy.
The night before they are due to harvest their Turnips, a member from the Green house decides to sneak out to ALL of the Turnip patches of the opposite family. They steal for every Green house family member from EVERY patch they visit. All of the other members of the Green house share a similar mind-set, and each individually, and unbeknownst of everyone else, decide to do the same thing.
Meanwhile, the Blue house get very greedy, and each occupant ends up individually stealing for every Blue house family member from EVERY Green Turnip patch.
In the morning, both families wake up to totally empty Turnip patches. Both family leaders go to the Turnip King to complain. The Turnip King decrees that all turnips must be returned to their original families. He gets all family members to convert their to and divide their amount EQUALLY between every family member of the opposite house. (This is done perfectly equally).
The Leader of the Green house complains to the King that every member of the Blue house has exactly more than everyone in the Green house.
The Turnip King has had enough of the rivalry and demands that all family members should join him for dinner to resolve their differences.
Including the chair for himself, what is the minimum amount of chairs required to accommodate everybody?
Tl;dr There are 2 groups of people Green and Blue.
Every Green member decides to individually steal 1 Gold for every Green members from every Blue member. At the same time, every Blue member decides to individually steal 13 Gold for every Blue members from every Green member. This leads to the original amount of Gold of both groups being perfectly stolen.
Every Green member has to divide their gold between all the Blue members. At the same time, every Blue member has to divide their gold between all the Green members.
Every Blue member has exactly 1 more gold than every Green member.
All group members and the king share a meal. What is the minimum amount of seats?
The answer is 830.
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1 solution
Let there be X members of the Green house, and Y members of the Blue house.
After the night of stealing, every Green house member has X Y G T , and every Blue house member has 1 3 X Y G T .
When dividing the turnips equally back to the opposite families, every Green member gives X G T each, and every Blue member gives 1 3 Y G T each.
Hence, every Green member ends up with 1 3 Y 2 G T and every member of the Blue house ends up with X 2 G T .
We are then told that X 2 − 1 3 Y 2 = 1 .
Over the positive integers (We know there is at least 1 member of each house), the smallest solution to this is:
X = 6 4 9 , Y = 1 8 0 , so the minimum number of seats is 6 4 9 + 1 8 0 + 1 = 8 3 0 .
This is a pell's equation. The smallest solution can be found by looking at the continued fraction of 1 3 = [ 3 ; 1 , 1 , 1 , 1 , 6 ] .
The 4 t h convergent of this gives Y X , where X 2 − 1 3 Y 2 = − 1 .
The 9 t h convergent of this gives Y X , where X 2 − 1 3 Y 2 = 1 :
[ 3 ; 1 , 1 , 1 , 1 , 6 , 1 , 1 , 1 , 1 ] = 3 + 1 + 1 + 1 + 1 + 6 + 1 + 1 + 1 + 1 1 1 1 1 1 1 1 1 1 = 1 + 1 + 1 + 1 + 6 + 5 3 1 1 1 1 1 = 1 8 0 6 4 9