July 5, 2001

**Old Challenge** (Nick Chung). While my wife was away, I invited n married couples to a party last weekend. An odd number of the guests were friends (acquaintances) of mine. Indeed, everyone had an odd number of friends at the party. What is the smallest possible value of n?

**Answer** (Raymond Hettinger, Carl Eichenlaub, Joseph DeVincentis, Branislav Kisacanin, Joshua Green ). There is no value of n for which this is possible. There are an odd number of people at the party, and each has an odd number of friends. Since the sum of an odd number of odd numbers is an odd number, the sum of all acquaintances per person must be odd. On the other hand, since every acquaintance involves two people, the sum of all acquaintances per person must be even. Since no number is both odd and even, the whole scenario is impossible.

**New Challenge** (Branislav Kisacanin). There are some positive numbers on my computer screen, and each of them is half the sum of the others. What could the numbers be?

*continued from last column.*

Last time we left our mathematician wandering around while waiting for the line at heaven's gate to get shorter.

**Scene II**

When he finally makes his way back to the Pearly Gates, the line is gone.

"There was some moving around inside, and uncountably many places opened up, and everyone went right in."

He mutters to himself: "Of course, why hadn't I thought of that. I hadn't really thought about rearranging uncountable sets before. It's quite interesting, actually." He is totally absorbed in his thoughts. By the time he looks up, another infinite line has formed.

"Oh no. Where do they all come from?"

"Surely you don't think that your little planet or what you call your universe is the whole show?"

"No, I guess not. I guess I'll get in line. How long will it be?"

"There's no way to tell. It varies from a few minutes to several thousand years. Rarely any longer than that."

He gets in line.

He happens to have a limit ordinal just five spots in front of him, just _{?} ?^{?} or something like that. In no time at all there are infinitely many people behind him, then a limit ordinal, then he can't see much farther. The soul in front of him is a nice chap from a universe which sounds very complicated topologically even on a relatively small scale. Half the culture seems devoted to keeping things untangled. The soul behind him is very happy, as if by habit, and does much to lift his spirits.

There is a small shift in heaven, and the first few limit ordinals get in, but of course that doesn't help him one bit. Finally after a week or two there is another, huge shift, and everyone is drawn into a waiting room in a tiny corner of the open space.

*To be continued in the next Math Chat.*

Copyright 2001, Frank Morgan.

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