## Decoding the mysterious symmetry of the bicycle lock numbers

Suppose you have a lock of this sort that has n dials and k numbers on each dial. Let m(nk) be the minimum number of turns that always suffice to open the lock from any starting position, where a turn consists of rotating any number of adjacent rings by one place.

In the previous post, we found an algorithm for computing these bicycle lock numbers, revealing a mysterious symmetry, Continue reading

## The bicycle lock problem

Photo: WickedVT

Don’t lock your bicycle with a combination lock. Someone will steal it: I learnt this the hard way. It’s quite easy to open a combination lock by feel, without knowing the combination. Try it: with a bit of practice, you can open the lock with your eyes shut. (It’s easier to do this with an old wobbly lock than a tight-fitting new one.)

Posted in algorithms, Mathematics | 4 Comments

## Beyond Bézier curves

There is a new feature of Pages and Keynote, not mentioned in any of Apple’s publicity nor in any press coverage I’ve seen, that is really very interesting. Perhaps it will even one day prove to have been revolutionary, in a quiet way. Continue reading

Posted in algorithms, Mathematics | Tagged , , | 8 Comments

## I hate the Pumping Lemma

I hate the Pumping Lemma for regular languages. It’s a complicated way to express an idea that is fundamentally very simple, and it isn’t even a very good way to prove that a language is not regular.

Here it is, in all its awful majesty: for every regular language L, there exists a positive whole number p such that every string w∈L that has p characters or more can be broken down into three substrings xyz, where y is not the empty string and the total length of xy is at most p, and for every natural number i the string xyiz is also in L.

Posted in chatter | 39 Comments

## The algebra of Unix command substitution

Shadab Ahmed raised an interesting question. Open a Unix command shell, type `: '!!'` and press return. Then type `: "!!" '!!'` and press return. Now repeat the following a few times: press the up arrow, and press return.

Posted in Mathematics | 6 Comments

## “Venn diagram” partitioning

Paddy3118 wrote about partitioning elements in the same way a Venn diagram does. So, if we have sets AB and C, the partitions are