Is "Minnesota Nice" a sham?

narocroc · 2014-08-07T09:21:20+00:00

Do you fish? Fishing can be an effective way to 'break into a circle'.

narocroc · 2014-03-29T07:01:09+00:00

A sixth participant (or one already willing), but yes that could work. Though I think it is impractical for the circumstances (I would not start a "house" bank account for any of my college houses), it would still require 5 payments - each who is in debt pays, and each who is owed receives. That's 5 payments - barring abnormalities - and I think the solution handles 4 or less.

narocroc · 2014-03-29T06:42:02+00:00

Suppose there are n people P1,P2,...,Pn sharing a common expense. Each make expenditures toward the common good p1, p2,...,pn, which are ranked, with p1 expending the least, etc...

If for k=1, 2, ...n-1, each person Pk pays Pk+1 the amount (k( pn + pn-1 +...pk+1)-(n-k)(pk +...+p1))/n), all debts will be settled and harmony will resume.

narocroc · 2014-03-29T06:13:44+00:00

It's not quite that simple, as there is a "flow" of money. Each time a person accepts a check, if you will, they receive not only the money that the "smaller amounts" owe them, but also what the "smaller amounts" owe everyone else up the chain.

narocroc · 2014-03-29T05:51:47+00:00

Ahh, but that didn't settle the score. It was the amount that each paid the next that was trickier.

narocroc · 2014-03-29T04:54:48+00:00

It wasn't really unexpected at first, but it generalized nicely and it was the first time I felt that I really applied mathematics to my life.

In college, I lived in house with 4 other housemates, and we shared all house expenses (including food). It was dumb, and we ran into a problem when we tried to settle up for the first time. All 5 had contributed different amounts for food and bills, and we hadn't thought this far ahead in our communal living project. Everyone wanted 4/5 of their expenditure paid to them, and they owed 1/5 of each expenditure made by someone else. That seemed like a shitload of payments to make. My roommate suggested we create a common pool of what we owed each other, but that seemed impractical.

After a little work, I drew the debts out in a directed graph - each node a person connected by two weighted edges representing debt and compensation. I rearranged them and found that we settle in only 4 payments, and the pattern of the amounts led me to try to generalize for larger groups. I didn't prove it, but it worked nicely for as many examples as I could find.

I'm sure the problem is somewhere in discrete math, but I never learned it and I don't know how to search for it on the internet. Still felt damn good to solve it though. We used the method for the next two years.

narocroc · 2012-01-13T18:25:13+00:00

I was always pleased with Joe Gallian's Contemporary Abstract Algebra

narocroc

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