29 thoughts on “Happy Birthday to me!

  1. One of the easiest mystery numbers is 153. Working in base ten (I clearly can’t put “10”), if you start with any positive integer divisible by 3, then keep forming new numbers by summing the cubes of their digits, you will eventually stop on 153.

    The proof is left as an exercise for the reader (I like saying that – that’s Schadenfreude for you).

  2. I’ll take the 30-second challenge on this one

    1. It’s trivial that 153 is a stopping point of the process
    2. You can easily get an upper bound from the fact that 9^3<1000
    3. Brute force enumeration of cases (left as an exercise for my computer) does the rest.

  3. You left out two important things, the difference between mathematics and number crunching.

    You should have done the computer stuff yourself, not just for the glory of it but because having your nose down in the dirt means you will spot oddities, things you would miss otherwise. Therefore it’s a good habit.

    And, sheer brute calculation will make you miss any patterns etc. that would reduce or remove the brute force. You completely missed the relevance of divisibility by three; that is invariant under the iteration (which you should prove), and eliminates 2/3 of the brute force cases needing checking.

    Mathematics works best when you can apply thought to cut back on work, which is to say work at a higher level.

    In the “real world”, whatever that is, you can be tricked by the elegance of the mathematics. But the right way to use mathematical models etc. is to get you most of the way there, both reducing the work (see above) and highlighting exceptions that don’t fit. That’s how these things advance best.

    I once did a brief essay on the distinctions between art, craft, and science, mostly to clarify it all in my own mind – its first version wasn’t very readable, even for me. But I think it’s useful to remain aware of the distinctions, since they can turn out very important. (Mathematics is not a science, by the way.)

Comments are closed.