First Thursday Maths Night

Here’s an idea for a great social occasion: invite a bunch of people around for the evening and solve maths problems.

You think I’m joking, don’t you!

There were 10 of us (I think (… I can count, I just can’t remember!)) at Anne and John’s last night, and we had a look at a few nice little puzzles. The problem that was intended as the warm-up exercise actually hijacked a goodly portion of the evening. It went something like this (names have been changed to avoid complications):

Harry is looking at Mary. Mary is looking at Jack. Harry is married. Jack is not. Is someone who is married looking at someone who is not? Is the answer “yes”, “no” or “you can’t tell for sure”?

Go away and think about it before reading on. In any case I’m not telling you the answer, so if you want to know you’ll have to think about it!

Being mathsy folk we wanted to think about generalisations of the problem. This is one of the reasons why we stayed on this problem for so long. Here are a couple of the generalisations we came up with (I think John invented the first; I did the second):

Harry is looking at Person-1. Person-1 is looking at Person-2. Person-2 is looking at Person-3. (… and so on …) Person-n is looking at Jack. Harry is married. Jack is not. Is someone who is married looking at someone who is not? Is the answer “yes”, “no” or “you can’t tell for sure”?

Hats can be three colours: red, blue or green. Everyone is wearing a hat. If Amy is looking at Bill and Cynthia, Bill is looking at Cynthia and Dirk, Cynthia is looking at Dirk and Amy, and Dirk is looking at Amy and Bill, then will anyone see someone wearing the same coloured hat as their own?

Being mathsy folk we wanted to think about isomorphisms of the first problem (an isomorphism of a thing is something that may appear to be different from the first thing at the superficial level, but in its heart of hearts, deep in its structure, it is identical).

One of my all-time favourite proofs in mathematics is the proof that it is possible to raise an irrational number to the power of an irrational and get an answer which is rational. (A rational number is just a fraction like 4/7 or -0.25; while an irrational number is one that can’t be written as a fraction, like √2 or π. Irrationals are characterised by the fact that if you write one as a decimal it goes on and on forever (like this parenthetical comment, seemingly) without having a recurring pattern, as in the number 0.10203040506070809010011012013014…. . All the numbers most of us know (i.e., the so-called “real numbers”), are either rational or irrational.)

Anyway, the proof about “irrational to the power of irrational can be rational” goes like this. Consider the number √2. This is one of our special irrational numbers (its decimal expansion is 1.41421356…. ). Now consider the number √2^√2. (Lucky I know enough html code to do that!). This is an example of an irrational number raised to the power of an irrational number. Now, I have no idea what the answer is! What on earth kind of number is √2^√2? However, it can only be one of two things. Either it’s rational or it’s irrational (since there are only these two types of real number). If it’s rational, then we’re done, because we found an example of an irrational to the power of an irrational being rational. On the other hand, what if it’s irrational? Well, if √2^√2 is irrational, then if we take this to the power of √2, i.e., (√2^√2)^√2 then this is an irrational being raised to an irrational power. But if you remember your index laws, you will know that (√2^√2)^√2 =√2^√2x√2 =√2² = 2, and since 2 is a nice lovely rational number, we’ve just found an example of an irrational to the power of an irrational being rational.

The really cool thing about this, is that we showed that an irrational to the power of an irrational can be rational without ever actually knowing what kind of number √2^√2 actually turns out to be.

Anyway, this turns out to be isomorphic to the married people looking at unmarried people problem.

Your mileage may vary of course. I, on the other hand, had a fun evening … and that’s even before we got to the “passing counters around” problem……..

[Just because I am anticipating that comments on this post may make reference to being rational or irrational, doesn’t mean you can’t still make them! Hints for the “married people looking at unmarried people problem” can be requested … but only after you’ve thought about it!]