Number of people n in a certain arrangement, find best move distance.
Given any number of people n, find a move
distance k and the minimum number
of moves of length k that creates a
clockwise sorted order;
Number n of people, one empty chair.
Generalize the problem with n people
and one empty chair to allow movements of distances k1, k2, …, kj for
some j < n/2; and
Several empty chairs. Generalize
the problem further to allow several
All are invited to submit solutions and prospective upstart-style
puzzles for future columns to email@example.com
Dennis Shasha ( firstname.lastname@example.org) is a professor
of computer science in the Computer Science Department
of the Courant Institute at New York University, New York,
as well as the chronicler of his good friend the omniheurist
Copyright held by author.
Solution to Warm-Up
C moves from 9 to 6
F moves from 3 to 9
C moves from 6 to 3
F moves from 9 to 6
Now here is a more difficult version
of the problem in which the only moves
allowed are four seats away, starting
with the configuration in Figure 2.
B moves from 6 to 2
F moves from 1 to 6
A moves from 5 to 1
E moves from 9 to 5
Now try to solve these four related
Number of people n and move distance k. Given any number of people
n and move distance k, find the minimum number of moves that would create a clockwise sorted order;
A GROUP OF people is sitting around your
dinner table with one empty chair. Each
person has a name that begins with a
different letter: A, B, C ... Because you
love puzzles, you ask them to rearrange
themselves to end up in alphabetical
order in a clockwise fashion, with one
empty chair just to the left of the person
whose name begins with A. Each move
involves one person from one chair to
the empty chair k seats away in either
direction. The goal is to minimize the
number of such moves.
Warm-up. Suppose you start with
eight people around the table, with
nine chairs. The last name of each person begins with the letter shown, and
you are allowed to move a person from
one chair to an empty chair three chairs
away in either direction (see Figure 1).
Can you do it in four moves?
Figure 1 (warm-up). The goal is to rearrange the people around the
table to be in clockwise alphabetical order, using as few moves as
possible, where each move involves moving a person three seats
away (in either direction) from the empty chair.
Figure 2. The goal is again to rearrange the people around the table
to be in clockwise alphabetical order, using as few moves as possible, where each move involves moving a person four seats away
(in either direction) from the empty chair.
DOI: 10.1145/2915924 Dennis Shasha