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June 5

Solitary, amicable, friendly

No, I'm not describing myself, here. In a mood to save the world, I set off to the unsolved problems and found that it is not yet known whether $10$ is a solitary number. According to Wikipedia, being solitary means not being part of an amicable pair. From the definition of that, however, it appears obvious that $10$ is not part of an amicable pair. Something seems to be wrong here, don't you think? Elsewhere, one seems to distinguish between amicable and friendly pairs, and define solitary as not being part of a friendly pair. Is that more correct? —Bromskloss 09:43, 5 June 2007 (UTC)[reply]

Yes, clearly 10 is not an amicable number or even a sociable number. My guess is that author of our solitary number article saw the definition that says a solitary number is a number that is not a friendly number (maybe here at MathWorld), saw that friendly number at Wikipedia redirects to amicable number, and then wrote the article bypassing the redirect, not realising that "friendly number" has two distinct meanings. I believe the open problem is whether or not 10 is a friendly number in the MathWorld sense i.e. whether there is any other integer n such that

\sigma (n)=1.8n

. Gandalf61 12:54, 5 June 2007 (UTC)[reply]

Does that make it an indifferent number? 213.48.15.234 13:01, 5 June 2007 (UTC)[reply]

OK, thanks. —Bromskloss 13:06, 5 June 2007 (UTC)[reply]

I have rewritten solitary number to follow the MathWorld definition. PrimeHunter 16:44, 5 June 2007 (UTC)[reply]

And I corrected the redirect at friendly pair to point to solitary number. Good catch. Dugwiki 16:47, 5 June 2007 (UTC)[reply]

And I've written a more substantial article Friendly number (replacing a remaining redirect to Amicable number) and redirected both Friendly pair and Solitary number to there. --Lambiam ^Talk 21:14, 5 June 2007 (UTC)[reply]

Rankings

I have a golf league in which I've ranked all the players 1 through 24.

What's a good formula to make teams of 4 with these 24 players as equal as possible. Currently I do something like the following:

A1, B4 A2, B3 A3, B2 A4, B1

but as far as 9-24, I'm at a loss.

You could try giving each person a numerical rating. Then ensure that the sums of the ratings for each of the six teams are equal. Zain Ebrahim 15:16, 5 June 2007 (UTC)p[reply]

I'm no statistician, but if you know the players' handicaps, you could work out the average handicap, and group them so that the sum of the groups' players' deviation from this mean sum to zero. This is just a more specific treatment of Zain's idea above. I can't instantly think of a situation where this would create a hugely unfair team...but I think this method doesn't account for something. Icthyos 16:21, 5 June 2007 (UTC)[reply]

Using just rankings, you could do like is done in some sporting events, match the highest to lowest, second highest to second lowest, etc. To get teams of four (not two which this naively suggests), break the pool of 24 into a top half, and a bottom half, and do each set independently. Then join a pair from each group in any way you like. One way would thus yield teams with rankings {1,12,13,24}, {2,11,14,23}, {3,10,15,22}, etc. You could also combine extreme pairs from one half with moderate pairs from the other: {1,12,18,19}, {2,11,17,20}, {3,10,16,21}, etc. The sum of the ranks for each team will be the same.

If you have more than just ranks, like the handicap example above, it may not be possible to get a subgroup whose deviations from the mean sum to zero (another way of saying the subgroup's mean and the whole group's mean are the same). You could try to minimize these differences over all six teams, but that is a linear programming problem and might require special software to compute. Baccyak4H (Yak!) 16:55, 5 June 2007 (UTC)[reply]

I believe this is a well-known problem in graph theory. The vertices represent the players and each edge indicates which player is better. The problem then becomes a kind of graph partition problem where we try to partition the vertices in such a way so that the partition weights are minimized. There are different ways to assign a weight to a partition, for example you could subtract the number of incoming edges from outgoing ones. Maybe someone here with a background in operations research can shed more light on this. nadav (talk) 20:26, 5 June 2007 (UTC)[reply]

Also closely related is 4-partition problem. nadav (talk) 21:05, 6 June 2007 (UTC)[reply]

statistics

how do i draw an ogive

Have you tried contacting an ogive modeling agency? —Tamfang 21:18, 5 June 2007 (UTC)[reply]

Could you be more specific? Do you want to draw just any ogive, or the ogive of a specific distribution? If the latter, how is that distribution given? By a formula? What are the drawing tools available to you? Pencil and paper? Matlab or Mathematica? --Lambiam ^Talk 21:44, 5 June 2007 (UTC)[reply]

simple, cubic, simplicial ...

What words are in most common use for an irregular polyhedron whose vertices are all of degree 3, or more generally, for a polytope whose vertex figures are all simplices? —Tamfang 21:18, 5 June 2007 (UTC)[reply]