Markov chains are used to model all kinds of different phenomena in nature; they're also intrinsically interesting and fun. One simple example of a Markov chain is a random process that generates a string of 1's, 2's, and 3's, where the only constraint is that no two consecutive digits can be equal;
after a 1, we're equally likely to see a 2 or a 3,
and after a 2, we're equally likely to see a 1 or a 3,
and after a 3, we're equally likely to see a 1 or a 2.

One way to generate such a string is to use three coins:
a penny (which we use after we see a 1; it has two sides, marked 2 and 3),
a nickel (which we use after we see a 2; it has two sides, marked 1 and 3),
and a dime (which we use after we see a 3; it has two sides, marked 1 and 2).

If we generate such a sequence using coin-flips to decide at each stage which of the two possibilities to append to the string, we get a sequence in which 1's, 2's, and 3's occur about equally often. In particular, if we generate N terms, the number of 1's, 2's and 3's will each be about N/3, with an error roughly on the order of sqrt(N).

To quasirandomize this procedure, rig the coins: if the LAST time you saw a 1 in the sequence it got followed by a 2, then the NEXT time you see a 1, follow it by a 3 instead! And vice versa. And likewise for the other cases. That is, instead of flipping the coin in the usual sense, just turn the coin over. If you do this, you get an infinite sequence like
that just consists of a finite string (in this case, 1,2,3,2,1,3) repeated over and over (possibly with some initial terms at the beginning that don't fit the pattern).

(If you're confused about the rule, here's an illustration of how we apply it: To figure out what comes next after that final "1", take a look back in the sequence until you find the PREVIOUS 1. Since that previous 1 was followed by a 2, the current 1 must be followed by a 3.)

Note that the infinite string 1,2,3,2,1,3,1,2,3,2,1,3,1,2,3,2,1,... has the property that if you look at just the part that repeats (1,2,3,2,1,3), you'll find that 1's and 2's and 3's don't just occur about equally often; they occur exactly equally often. So, if you stopped the game after N terms, you might not get each of the three values to occur exactly N/3 times, but the error will be bounded (instead of growing with N the way sqrt(N) does).

That's a typical sort of situation for quasirandomization: if you take out the randomness and replace it by something that's non-random but "fair", then you get a non-random system that reproduces some (though not all!) of the features of the random system that you're trying to study, often in a more exact way.

The $100,000 question is, does this extend to the sorts of random processes that people other than probability theorists care about? (That dollar amount isn't random, by the way; it's the total budget for my National Science Foundation grant to study quasirandomness.)

A much more interesting example (and one that researchers have barely begun to understand) is quasirandom aggregation, which gives cool pictures like You can learn where that picture comes from by reading Michael Kleber's nice article.

The study of quasirandomness pretty much runs the gamut between these two examples: one nearly trivial, the other very deep. In my Fall 2007 probability course, I'll start students off on the shallow end; by the time EQL finishes its work in summer 2009, I hope we'll have made some progress towards the deep end.

If you're interested in finding out more, send me email! Write to (remove everything from my email address after the "edu"). You can also browse through what's at

Jim Propp