## 18 March 2011

### What is the origin of Kirillov's lucky number problem?

I've run across a few references to a Russian superstition as follows: a bus ticket has a six-digit number, and a ticket is said to be "lucky" if the sum of the first three digits equals the sum of the last three digits. See, for example, Lectures on Generating Functionsby Sergei Lando -- an excellent little book I accidentally discovered in the library a couple days ago. Lando gives the problem of figuring out how many lucky tickets there are, and says that in the early 1970s A. A. Kirillov would often open his seminar this way. The cataloging information in the book says that Lando was born in 1955; Olshanski writes, in his preface to Kirillov's seminar on representation theory, that many students attended Kirillov's seminar. I suspect that Lando actually heard Kirillov pose this problem at some point.

What I'm interested in is whether this is an actual superstition, perhaps held by non-mathematicians. It seems like the sort of thing that a mathematician would make up to create a good problem. This web page mentions the superstition in a non-mathematical context, with the twist that you're supposed to eat lucky tickets. This one does as well, and links to these people who sell lucky ticket cookies. It seems quite likely that:

1. the superstition exists among some segment of the Russian population, or at least did at some point, and
2. the entrance of the problem into the mathematical culture is due to Kirillov -- if I were still at Penn I'd ask him.

But did the problem travel from the rest of the world to mathematics? Or, because this sum condition feels so mathematical, did it travel from mathematics to the rest of the world?

(Also, can you solve the problem? How many lucky tickets are there?)

J.T. said...

55252?

Dan said...

J.T.: that's what I got, both by an exhaustive search in Sage and by squaring the coefficients of (1+x+x^2+...+x^9)^3.

Michael Lugo said...

That's the right answer. I haven't done the exhaustive search but I can confirm that's what you get by squaring the coefficients.

jd2718 said...

Is 000000 a ticket number? Is 000001 a ticket number?

Jonathan

Aniko said...

I went to grade school in Russia (Moscow) in the 80s, and the concept of "lucky tickets" was absolutely around among quite ordinary schoolchildren. And yes, you were supposed to eat them.

apn said...

It also equals the number of six digit numbers with a digit sum 27.