Knight's Tour
Actually there is no need for backtracking to achieve a completed knight's
tour. There is an old efficient method that goes straight to that goal.
This is an old problem dealt with already by Leonard Euler (1707-83). The
question raised is, whether the chess piece the knight could make a tour
around the board, thereby visiting every square once and just once. And if
there is such a tour, immediately another question emerges. How many
tours are there?
In the beginning of the 19th century a practical method for solving the
Knight's tour emerged. In "Des Rösselsprungs einfachste und allgemeinste
Lösung" (Schmalkalden, 1823) H. C. Warnsdorff presented his method of
constructing knight's tours. The aim is to avoid creating dead ends - squares
from which the knight cannot get further without getting to an already
visited square. For that reason the possible squares to be chosen next are
examined before every move. One counts the number of free new choices entrances - every one of them has, and then moves to the square with the
lowest number of new choices. Warnsdorff's rule is heuristic.
The problem can be restricted by demanding the tour to be closed (or reentrant): from the last square in the tour the knight should be able to reach
the starting square.
One might think that the problem ought to be solved nowadays when we
have got computers. An easy way of solving it seems to be to let the
computer test all possible ways of choosing ways the knight has. Most
choices will result in a dead end situation - the knight gets to a square from
which it cannot get away without getting to a square already visited. In that
situation just go back one step, and choose another square to go to. And if
you have already tried all possible choices from that square, just go back
another step. Backtracking is the name of that strategy, and it is often
applicable
The problem is not too difficult on a smaller board. On a 6*6 squares board
simple backtracking is a sufficient strategy for counting re-entrant tours. On
such a board there are 9 862 different closed tours, a result obtained after
testing 4 056 367 434 moves. (Although backtracking always guarantees a
result - if you've got the time - it can be rather inefficient)
On a normal 8*8 squares board the problem turns out to be surprisingly big.
Actually so big that simple counting of tours is out of reach even for the fast
computers of today. The problem has to be tackled in other ways.
In 1995 Martin Löbbing and Ingo Wegener wrote a paper with the appealing
title: "The Number of Knight's Tours equals 33,439,123,484,294 -- Counting
with Binary Decision Diagrams". They obtained their result by running 20 Sun
stations for four months (idle time, efficient CPU time being less). Brendan
McKay issued a comment on 18 Feb. 1997. He used another method, and
split the board into two halves, obtaining the result 13,267,364,410,532.
To give an idea of the magnitude of these numbers, consider a computer
searching and finding tours at a speed of 1 million tours per minute (266
MHz PC). This may seem pretty fast, but the computer will nevertheless
have to run for more than 9,213 days and nights - i.e. for more than 25
years - to reach the number of tours given by Brendan McKay.
Theoretically there are objections, but on a normal 8*8 squares board the
rule works just fine. Example: Take a look at a square in a corner of the
board. From such a square a knight can jump to only two other squares. And
the square in the corner can be reached only from these two other squares the square in the corner has two "entrances". When a knight during a tour
happens to visit any of these entrances, it is almost bound to visit the
corner square next. If the knight doesn't do that, it has used up one of the
two free entrances of the corner square - and thus turned that square into a
dead end; the corner square can still be reached later, but there will no
longer be any unused way out.
During a knight's tour the number of free entrances to all squares of the
board are gradually used up. Warnsdorff's rule serves to direct the knight to
the squares with the least numbers of free entrances left - before these
squares have turned into dead ends.
Warnsdorff's rule gives solutions, but not all possible solutions (One can
make moves opposing the rule and yet get a complete tour). The rule
possesses a trait of arbitrariness; there is often a choice between equal
alternatives. And on really big boards the rule runs into trouble.
Find words corresponding to the definitions:
Give the English for: (some of them are in the text)
Intended to increase the probability of solving some problem by using common sense
Case
Echecs
Echec et mat
Echiquier
Pat
Plateau
Rangée
Roquer
To break apart
To set about dealing with
The state of acting more from caprice than from reason or judgment
By that means or because of that
Find the English for:
Write the questions corresponding to the underlined elements in the following
sentences taken from the text:
Impasse
Pas
Certain, sûr
Assez, relativement
Efficace
This is an old problem dealt with already by Leonard Euler.
Find the opposites:
The problem turns out to be surprisingly big.
Immediately
Repulsive
Confront
In the beginning of the 19th century a practical method for solving the
Knight's tour emerged.
Complete the table:
English
r
p
k
n
q
b
Français
Warnsdorff's rule is heuristic.
From such a square a knight can jump to only two other squares.
Phrasal verbs:
Find the phrasal verbs corresponding to the following definitions using GET or
TURN as the stem, and one of the following prepositions :
What do these commonly used abbreviations mean?
ALONG, AWAY, BACK, DOWN, IN, INTO, OFF, ON,
OUT OF, OVER, UP
Arise from bed, a chair etc.
Decrease volume OR refuse
enter a bus or train etc. OR have a good relationship with
enter a car
escape
Extinguish a light OR appear, happen
have a good relationship with
Increase volume or intensity
Leave a bus or train etc.
leave a car OR avoid some unpleasant activity
Latin
Meaning
A.M.
c. or ca
cf.
e.g.
et al.
etc.
i.e.
N.B.
op. cit.
P.M.
P.S.
Q.E.D.
v., vs.
viz.
Observe the uses of ‘so’ and ‘such’ in the text. Can you determine a rule?
Complete and translate:
recover from an illness or a bad experience
My processor is _____________ hot today. I hope the fan hasn’t stopped working.
return from somewhere OR retrieve
Reverse
Start a machine, equipment, light etc. OR arouse sexual interest
Stop a machine, equipment, light etc.
submit classwork OR go to bed
transform
He’s written ____________ a useful program to calculate moon phases.
Windows crashes ____________ frequently that I’m thinking of switching to Linux.
Nowadays many people use instant messaging, but recent events have shown that
_______________ programs are vulnerable to Trojan horse viruses.
I’ve installed __________ many different browsers on my computer that I don’t
know which one to use.