# Word problem (food for thought)

I thought of this question today as I was coming home from work in my car (probably because of my parents' anniversary).

This problem assumes the parents of everyone in the world got married and everyone in the future gets married at some point in time (a.k.a. it's hypothetical). If every couple planned their wedding day to occur exactly halfway between the dates of the anniversary of their parents' weddings, would everyone eventually have the same wedding day?

To illustrate this problem more clearly:

$P_m=$ date of wedding for parents of the male

$P_f =$ date of wedding for parents of the female

$C = \frac {(P_m + P_f)}{2} =$ date of future wedding for couple

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What does 'halfway' mean here? Or, equivalently: if one set of parents got married on December 31st, and another on January 2nd, are those dates 2 days apart or almost a year apart? Do their children get married at the start of January or at the start of July? –  Billy Jun 28 '13 at 0:45

Not necessarily. Suppose everyone married a person of the same nationality as themselves. You could have all marriages in England occurring on 5 June, in Australia on 12 November etc.

Create a model in which every month has two days, and the first round of weddings consist of a single wedding on the first of each month. Each couple has two children. The first child born to the March couple marries a February child, the second March child marries an April child. The weddings all happen on the second of the month, and they all have two children. If the Dec/Jan wedding happens on 2 Dec (see Billy's comment) then you have a system where the weddings in each generation swap between the first and the second of the month.

If the Jan/Dec wedding happened on 2 June, there could be no future weddings on 1 Jan or 2 Dec. Depending what happens to $C$ when $P_m+P_f$ is odd, you can see that whenever a person who belongs to the earliest date or the latest date marries someone who belongs to a date at least two days around, (and depending on the relative birth rates for couples belonging to the different dates) the proportion of the population attached to the extreme dates will tend to fall, and the range of dates on which marriages take place will narrow.

But for a fully specified problem you need to pin down the answer to Billy's question, how to deal with dividing odd numbers by 2 (it would be possible on some scenarios - eg resolving in the direction of the man - to have a stable system with marriages taking place on the same two adjacent dates in each generation) and also to say something about mixing relative to birthrate.

Even in a system where people are mixing, if the mixing rate is low and the birth rate at extreme dates is high, there can be a persistence of weddings at the earliest and latest date rather than a narrowing of the date range.

So that illustrates a few of the possibilities and issues.

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+1 nice example that the convergence is local, not global. –  Calvin Lin Jun 28 '13 at 2:27
I have added some more reflection to highlight some other issues - the question is more complex than it seems. –  Mark Bennet Jun 28 '13 at 6:28
That's a great way of thinking about it. I would have never thought the dates could diverge. To answer the question by Billy, I would think if the marriages are at extremes (early January and late December), the resulting marriage would occur in July. For dividing odd numbers, I suppose rounding up would make most sense because you would get 0.5 as a remainder. –  Jason Jun 29 '13 at 1:30