## Saturday, April 9, 2016

### Alphabet Slider, Cont'd

The slider mechanism in the Alphabet Slider puzzle determines what's called a "partial ordering" on three-letter words. Specifically, given two words w1 and w2, we write

w1 < w2

if every letter of w1 occurs earlier in the alphabet than the corresponding letter of w2. For example, we write

ACE < PYX

because A is earlier in the alphabet than P, C is earlier in the alphabet than Y, and E is earlier in the alphabet than X.

There exist words "between" ACE and PYX. For example,

ACE < EON < PYX

as you can check. A solution to the Alphabet Slider is a chain like the one above, and the goal is to find the longest possible chain. With the help of a computer, I found this solution:

ACE < BEG < CHI < DIM < EON < FRO < ITS < JUT < PYX.

(An equally long chain results from using NUT instead of JUT. A reader also notes that OUT could go in this position.)

Here's a picture—note that lines never cross.

The game can be played with words of any length. For example, here's a pretty good chain of four-letter words:

ABBE < DILL < ELMS < FONT < GROW < JURY.

For eight-letter words, I found