The Josephus Problem
"What one fool can do, another can."
This chapter of Concrete Mathematics is trickier than the previous ones. I drew many images and found the pattern myself but had to turn to the book for advice on how to formalize it.
And… Anti-Congrats to myself for bumping into the first derivation I'm not getting 😕 . Or rather it should be called a gap in the derivation. I noticed that mathematicians like those gaps so much. As a programmer 👩💻 , I'm used to writing down the instructions precisely, one by one, without omitting any steps required for the successful execution of a task at hand. It's different with mathematicians. They like to skip steps. (IMO, one of the rare exceptions to this tendency is the book Discrete Mathematics with Applications written by Susanna S. Epp.)
Supposedly, it means the authors expect their readers should be able to grasp the gaps left. Or the authors don't notice there are any gaps at all. Or what if they do it on purpose, to set a high bar for people trying to understand the topic? This might make sense. The fewer people get mathematics, the more mathematicians are in deficit; with the consequence that math professionals are in greater demand and self-perceived value. Well, I really don't know. I'm still figuring out the answer to this question 1.
So now I'm going to take a break, probably for a day or two. Such breaks often help to "digest" a task to solve, so I hope the next time I delve into this topic again I'll be able to fully understand the derivation and fill in the gaps.
Footnotes:
Appears I am not alone in my perception. The following is
the quote from the — still popular! — book Calculus Made
Easy by S. P. Thompson, first published back in 1910:
Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.
Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics — and they are mostly clever fools — seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.
Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.
