## The “rho” – A Mathematical Diversion

### Mathematics, Strategy

July 27, 2007

Got nothing better to do? Here’s something to keep you entertained. I promise this will get to the subject of poker strategy in a future post, but for now I’m hoping your love of mathematics will keep you riveted. This works OK as a thought experiment, but if you’ve got time on your hands I suggest you actually try it. Here we go!

Take a standard piece of lined paper and number it down the left side starting at 1 and going until you run out of lines. Then fold it vertically (or as my 1st grade teacher used to say, like a hotdog) a couple of times to create a number of columns. Four is about right. Eight is probably too many unless you’re really bored. Now, in the seccond column, on each line write one number between 1 and the highest number you got to in the first column. You can select which number to put on a given line by any means you like as long as it’s in that range. You can pick as randomly as you can manage (or roll dice), use every number once, put ‘7’ on each line or put the numbers in reverse order. Whatever floats you boat.

Now repeat this process in however many columns you have left. In each column, choose some different scheme for assigning numbers to the lines.

Now, here’s the experiment. The left column contains line numbers. Pick one line (doesn’t matter which), put your pencil there, and put a little tick mark there to indicate you’ve been to that line. Then look in column two on that line, treat the number you see there as a line number, and move your pencil to that line. Mark it off and repeat the process.

What happens? Eventually you end up going in circles. You’ll know this because you get back to a tick mark. At that point you can loop around for a while, but it should be pretty clear that you’re not going to visit any more numbers, just the same ones over and over again. When you’re convinced of this, stop. You’ve encountered something called a “cycle”.

Now erase your tic marks and try again. Instead of using column two, you can use another one you prepared (you’ll always use column one as the line numbers), start on a different line, etc. to get some variety and excitement in your life.

Now one more little experiment: get another piece of paper. As you step through the lines, for each line draw a small circle with the line number written in it and connect the circles with arrows as you progress. When you get to the step that goes back to a line you’ve been to, draw an arrow back to that circle instead of creating a new one with the same number. At that point you’re done. You should end up with something that looks like an overly mathematical sperm.

So what’s the point here? Several things:

• Each an every time you do this, you get stuck in a cycle. It’s unavoidable. This is a direct consequence of the pigeonhole principle with steps being pigeons and lines being holes.
• When you end up back at a line you’ve been to before, it’s usually not the line you started on. This is obvious when you draw the structure out on paper – you get a “tail” that’s not in the cycle, and a “head” that is.
• There are two degenerate structures – a head with no tail which can be achieved by putting “2” on line 1, “3” on line 2 and so on and “1” on line N and a head with only one element that loops back on itself, which can be created by putting a number on it’s own line.

These structures, drawn out, are sometime referred to as “rhos” because of their visual similarity to the Greek letter of the same name. They have something to do with poker strategy too, and I’ll get to that in my next post.