In the first story, Lucky Number, a game programmer spots some "lucky numbers" spray painted on a train. On a whim, he uses them as the rule for a new cellular automaton. Cellular automata are a subject of much mathematical research; informally you could say that each one is a finite dimensional discrete universe in which we set the "laws of physics" and see what happens. As it turns out, there are many interesting possible "universes" that can be constructed in this way. Perhaps the most famous is Conway's "Game of Life". The complexity of some of these systems have lead some to suggest that the real universe (the one in which we live) might itself be a cellular automaton. (IMHO, this is not a ridiculous suggestion. But, it is ridiculous to take it too far. Since cellular automata do display great complexity, it is possible that something like the real universe could be a result of the right set of rules and the right initial condition. However, there are lots of different sorts of math that lead to complex results. For instance, differential equations also describe rather complex situations. There is no reason -- at this point -- to assume that the universe is best described as a cellular automaton as opposed to any other kind of math.) The story rolls with this idea, so that after the character uses the lucky numbers as a set of rules, the cellular automaton quickly runs through the history of the universe (including dinosaurs and Jesus). Then, as if that wasn't enough, there is one final twist. [One final note on cellular automata: IMHO, although Steve Wolfram is one of many researchers who has made contributions to this field of mathematics, he does have a tendency to overemphasize his own role and the significance of this topic. If you read his book or other writings on the subject, approach such claims with skepticism.]

The second story mixes Buddhism with quantum physics to allow a woman to afford an apartment in California despite high rents...but is not very mathematical.

The third story is only mathematical in the sense that it introduces the idea of a paint that uses activator-inhibitor processes like the Belousov-Zhabotinsky reaction (unfortunately misspelled in the story) to generate interesting patterns. (That is chemistry as much as it is mathematics, but such reactions can be studied mathematically as well.)

There's no math at all in the fourth story (about a man who builds a machine to hold conversations for him, like the programs often displayed by researchers in artificial intelligence). Similarly, the fifth story -- about a woman who can see patterns in the rain -- is not really mathematical either.

Finally, Hello Infinity is quite mathematical, and reminiscent of the ideas in Rucker's first novel, White Light. An accountant on his day off discovers that he has the ability to count to infinity (by visualizing each number in half the time it takes for the one before). Coincidentally, his scientist wife comes up with a new idea for a microscope (using octopus skin for the display!) on the same day. Putting the two together, they can see the infinitessimal...and consider quitting their jobs.

This story was first published in Lifebox, the Seashell and the Soul and was reprinted just a year later in Mad Professor. As of October 2012, the story can also be freely downloaded from the author's Website.