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Frequently Asked Questions (FAQS);faqs.464
==> games/connect.four.p <==
Is there a winning strategy for Connect Four?
==> games/connect.four.s <==
An AI program has solved Connect Four for the standard 7 x 6 board.
The conclusion: White wins, was confirmed by the brute force check made by
James D. Allen, which has been published in rec.games.programmer.
The program called VICTOR consists of a pure knowledge-based evaluation
function which can give three values to a position:
1 won by white,
0 still unclear.
-1 at least a draw for Black,
This evaluation function is based on 9 strategic rules concerning the game,
which all nine have been (mathematically) proven to be correct.
This means that a claim made about the game-theoretical value of a position
by VICTOR, is correct, although no search tree is built.
If the result 1 or -1 is given, the program outputs a set of rules applied,
indicating the way the result can be achieved.
This way one evaluation can be used to play the game to the end without any
extra calculation (unless the position was still unclear, of course).
Using the evaluation function alone, it has been shown that Black can at least
draw the game on any 6 x (2n) board. VICTOR found an easy strategy for
these boardsizes, which can be taught to anyone within 5 minutes. Nevertheless,
this strategy had not been encountered before by any humans, as far as I know.
For 7 x (2n) boards a similar strategy was found, in case White does not
start the game in the middle column. In these cases Black can therefore at
least draw the game.
Furthermore, VICTOR needed only to check a few dozen positions to show
that Black can at least draw the game on the 7 x 4 board.
Evaluation of a position on a 7 x 4 or 7 x 6 board costs between 0.01 and 10
CPU seconds on a Sun4.
For the 7 x 6 board too many positions were unclear. For that reason a
combination of Conspiracy-Number Search and Depth First Search was used
to determine the game-theoretical value. This took several hundreds of hours
on a Sun4.
The main reason for the large amount of search needed, was the fact that in
many variations, the win for White was very difficult to achieve.
This caused many positions to be unclear for the evaluation function.
Using the results of the search, a database will be constructed
of roughly 500.000 positions with their game-theoretical value.
Using this datebase, VICTOR can play against humans or other programs,
winning all the time (playing White). The average move takes less
than a second of calculation (search in the database or evaluation
of the position by the evaluation function).
Some variations are given below (columns and rows are numbered as is customary
in chess):
1. d1, .. The only winning move.
After 1. .., a1 wins 2. e1. Other second moves for White has not been
checked yet.
After 1. .., b1 wins 2. f1. Other second moves for White has not been
checked yet.
After 1. .., c1 wins 2. f1. Only 2 g1 has not been checked yet. All other
second moves for White give Black at least a draw.
After 1. .., d2 wins 2. d3. All other second moves for White give black
at least a draw.
A nice example of the difficulty White has to win:
1. d1, d2
2. d3, d4
3. d5, b1
4. b2!
The first three moves for White are forced, while alternatives at the
fourth moves of White are not checked yet.
A variation which took much time to check and eventually turned out
to be at least a draw for Black, was:
1. d1, c1
2. c2?, .. f1 wins, while c2 does not.
2. .., c3 Only move which gives Black the draw.
3. c4, .. White's best chance.
3. .., g1!! Only 3 .., d2 has not been checked completely, while all
other third moves for Black have been shown to lose.
The project has been described in my 'doctoraalscriptie' (Master thesis)
which has been supervised by Prof.Dr H.J. van den Herik of the
Rijksuniversiteit Limburg (The Netherlands).
I will give more details if requested.
Victor Allis.
Vrije Universiteit van Amsterdam.
The Netherlands.
victor@cs.vu.nl
==> games/craps.p <==
What are the odds in craps?
==> games/craps.s <==
The game of craps:
There is a person who rolls the two dice, and then there is the house.
1) On the first roll, if a 7 or 11 comes up, the roller wins.
If a 2, 3, or 12 comes up the house wins.
Anything else is a POINT, and more rolling is necessary, as per rule 2.
2) If a POINT appears on the first roll, keep rolling the dice.
At each roll, if the POINT appears again, the roller wins.
At each roll, if a 7 comes up, the house wins.
Keep rolling until the POINT or a 7 comes up.
Then there are the players, and they are allowed to place their bets with
either the roller or with the house.
-----
My computations:
On the first roll, P.roller.trial(1) = 2/9, and P.house.trial(1) = 1/9.
Let P(x) stand for the probability of a 4,5,6,8,9,10 appearing.
Then on the second and onwards rolls, the probability is:
Roller:
--- (i - 2)
P.roller.trial(i) = \ P(x) * ((5/6 - P(x)) * P(x)
(i > 1) /
---
x = 4,5,6,8,9,10
House:
--- (i - 2)
P.house.trial(i) = \ P(x) * ((5/6 - P(x)) * 1/6
(i > 1) /
---
x = 4,5,6,8,9,10
Reasoning (roller): For the roller to win on the ith trial, a POINT
should have appeared on the first trial (the first P(x) term), and the
same POINT should appear on the ith trial (the last P(x) term). All the in
between trials should come up with a number other than 7 or the POINT
(hence the (5/6 - P(x)) term).
Similar reasoning holds for the house.
The numbers are:
P.roller.trial(i) (i > 1) =
(i-1) (i-1) (i-1)
1/72 * (27/36) + 2/81 * (26/36) + 25/648 * (25/36)
P.house.trial(i) (i > 1) =
(i-1) (i-1) (i-1)
2/72 * (27/36) + 3/81 * (26/36) + 30/648 * (25/36)
-------------------------------------------------
The total probability comes to:
P.roller = 2/9 + (1/18 + 4/45 + 25/198) = 0.4929292929292929..
P.house = 1/9 + (1/9 + 2/15 + 15/99) = 0.5070707070707070..
which is not even.
===========================================================================
==
Avinash Chopde (with standard disclaimer)
abc@unhcs.unh.edu, abc@unh.unh.edu {.....}!uunet!unh!abc
==> games/crosswords/cryptic/clues.p <==
What are some clues (indicators) used in cryptics?
Xref: bloom-picayune.mit.edu rec.puzzles:18144 news.answers:3075
Newsgroups: rec.puzzles,news.answers
Path: bloom-picayune.mit.edu!enterpoop.mit.edu!snorkelwacker.mit.edu!usc!wupost!darwin.sura.net!haven.umd.edu!uunet!questrel!chris
From: uunet!questrel!chris (Chris Cole)
Subject: rec.puzzles FAQ, part 8 of 15
Message-ID: <puzzles-faq-8_717034101@questrel.com>
Followup-To: rec.puzzles
Summary: This posting contains a list of
Frequently Asked Questions (and their answers).
It should be read by anyone who wishes to
post to the rec.puzzles newsgroup.
Sender: chris@questrel.com (Chris Cole)
Reply-To: uunet!questrel!faql-comment
Organization: Questrel, Inc.
References: <puzzles-faq-1_717034101@questrel.com>
Date: Mon, 21 Sep 1992 00:09:26 GMT
Approved: news-answers-request@MIT.Edu
Expires: Sat, 3 Apr 1993 00:08:21 GMT
Lines: 2382
Archive-name: puzzles-faq/part08
Last-modified: 1992/09/20
Version: 3
==> games/crosswords/cryptic/clues.s <==
The following list is derived from indicators used in a variety of
crosswords: the letters in the left column are the letters being
indicated; the right hand column is how these letters might be
indicated in a clue.
Caveat emptor: many of the entries in this list would be considered
unsound in some puzzles (some of these unsound indicators are marked
with a +). Entries marked * are used mostly in advanced cryptics.
I would welcome corrections and additions to the list.
----------------------------------------------------------------------
a Austria
a I
a academician
a accepted
a ace
a acre
a active
a adult
a advanced
a afternoon
a aleph
a alpha
a amateur
a ampere
a an/ane
a angstrom
a answer
a ante
a are (metric)
a articles - English
a associate
a atomic
a ay
a aye
a before
a blood group
a bomb
a effect
a examination
a fifty
a film
a first character
a first class
a first letter
a five hundred
a five thousand
a good
a high class
a it
a key +
a level
a midday
a note +
a one *
a paper
a road
a strings (violin)
a un
a unit
a violin string
a vitamin
a year
aa motoring organisation
ab able seaman
ab hand
ab rating
ab sailor/salt/seaman
ab tar
abbe priest (Fr.)
abe Lincoln
abel first victim
abel murder victim
abel second child
abel third man
able can
able expert
abo native
ac account
ac accountant
ac aircraftsman
ac alternating current
ac before Christ
ac bill
ac current
aca accountant
acas peacemakers
acc account
acc bill
ace card
ace champion
ace expert
ace one
ace pilot
ace service
ace winner
act decree
act performance
actor tree
ad Christian era
ad advertisement
ad after date
ad before the day
ad contemporary
ad in the modern age
ad in the year of our Lord
ad modern times
ad notice
ad now
ad nowadays
ad our time/era
ad period
ad present day
ad promotion
ad puff
ad this era
ad today
adam first character
adam first person
adam number one
add sum
add tot
aden port
admin management
admin running
ado business
ado difficulty
ado fuss
ado row
ado trouble
ae aged
ae poet
aet aged
ag silver
aga Muslim leader
age (long) time
age mature
age period
agent spy
agm annual meeting
agm meeting
agm yearly meeting
ai capital
ai first class
ai good
ai high class
ai main road
ai sloth
ail trouble
ain own (Scot.)
air appearance
air display
air song
aire river
ait island
al Alabama
al Alan
al Albania
al Albert
al Capone
al aluminium
al gangster
al one pound
ala Alabama
ala after the style of
ala in the style of
ala to the (Fr.)
ala wings
alas Alaska
alb one pound
ale beer
aleph Hebrew letter
all completely
all everybody
all everything
alp mountain
alp peak
alph river
alpha Greek letter
alpha beginning
alpha first character
alpha first letter
am America
am American
am I am
am admitting
am boasting
am half day
am hymns
am in the morning
am morning
am self-confessed
amen final word
amen last word
amer American
ammo missiles
amos bookmaker
amp one member
an I
an articles - English
an before
an if (old word)
an one *
an un
ana tales
ane I
ane one
ane one
ankh life symbol
anne princess
anon now
ans answer
ans brief reply
ans collection
ans short answer
ant if it (old word)
ant six-footer
ant social worker
ant soldier
ant worker
ape copy
ape primate
aq water
ar arrive/arrival
ar year of reign
ara academician
ara artist
ara painter
arab horse
arc curve
argo old ship
aria song
arm gun
arm limb
arm member
arr arrive/arrival
art contrivance
art craft
art cunning
art painting
art skill
as Anglo-Saxon
as ayes
as ays
as like
as one's
as specifically
as when
ash remains
ash tree
asia continent
aside one fifteen
asis existing state
asp snake
ass donkey
asti wine
ate goddess
ate mischief
athena goddess
ation at one on
atoll bikini
au gold
au to the (Fr.)
aus Australia
aux to the (Fr.)
av bible
av lived so long
ave average
ave greeting
ave hail
ave road
ave way
aver average
avon county
ay I
ay agreement
ay always
ay ever
ay yes
aye I
aye I say
aye agreement
aye always
aye ever
aye yes
az Azed
az scope, plenty of
b Bach
b Beethoven
b Belgium
b Brahms
b Britain
b British
b a follower
b bachelor
b baron
b bedbug
b bee
b bel
b beta
b beth
b bishop
b black
b blood group
b bloody
b book *
b born
b boron
b bowled
b boy
b breadth
b inferior
b key +
b magnetic flux
b note +
b paper
b second
b second class
b second letter
b three hundred
b three thousand
b vitamin
ba Bachelor of Arts
ba airline
ba bachelor
ba barium
ba degree
ba graduate
ba scholar
bac airline
bacon philosopher
ban curse
ban outlaw
ban prohibition
bar Inn
bar lawyers
bar prevent
bar save
barb horse
bat fly-by-night
bb bees
bb books
bb very black
bc ancient times
bc before Christ
bc period
bd beady
bd bound
bd cleric
bd theologian
be exist
be live
bea airline
bear speculator
bed in bed
bee buzzer
bee group of workers
bee six-footer
bee social worker
bee worker
bef Gort's men
bess queen
beta Greek letter
beth Hebrew letter
bi double
bi two (double)
bird prison
bis two (twice)
bit chewed
bit piece
biz business
bk book
bl British company
bl lawman
bl lawyer
blue Conservative
bm British Museum
bm doctor
bo American man
boa snake
board directors
bob old shilling
bp bishop
br Britain
br British
br British Rail
br bank rate
br branch
br bridge
br brig
br brother
br brown *
br lines/landline
br railway(s)
br trains
br transport
bra female support(er)
bra support
bra undergarment
brass money
brat child
bren gun
brer rabbit
bridal old wedding
bro brother
bs bees
bst summer time
bull American policeman
bull gold
bus transport
c Celsius/centigrade
c Charles
c Conservative
c Cuba
c about (approx.)
c approx(imately)
c around
c cape
c caput
c carbon
c caught
c cedi
c cent/centime
c centi-
c century
c chapter
c circa
c club
c cold
c complex number
c copyright
c coulomb
c electrical capacity
c hundred
c hundred thousand
c key +
c lot
c many
c note +
c roughly
c sea
c see
c speed of light
c spring
c tap
c vitamin
ca about (approx.)
ca accountant
ca approx(imately)
ca calcium
ca roughly
cab transport
cade conspirator
cain first murderer
cain killer
cain murderer
cal California
cam river
can able to +
can is able to
can prison
can vessel
cantuar archbishop
cap chapter
cap international
car carat
car transport
carnation motor race
cart transport
cat jazz fan
cato censor
cattle neat
cave warning
cb Seabee (Amer.)
cc county council
cc seas
cc small measure
cc small quantity
cc two hundred
cd diplomat
cd seedy
ce Church of England
ce church
ce engineers
ce this (Fr.)
cent money
cet this (Fr.)
ch China
ch Companion of Honour
ch Switzerland
ch award
ch central heating
ch champion
ch chapter
ch chief
ch child
ch church
ch companion
ch honour
ch order
cha tea
chai gypsy woman
chair president
chal gypsy
char daily
che guerrilla
che revolutionary
cher dear (Fr.)
chere dear (Fr.)
chi Greek letter
ci Channel Islands
ci hundred and one
cia secret service
cia spies
cid captain
cid chief
cid detectives
cid police
cid spanish hero
cinc commander
cl chlorine
cl class
cl clause
cl gas - chlorine
cl hundred and fifty
co Colombia
co business
co care of
co cobalt
co commander
co commanding officer
co company
co county
co firm
co gas - carbon monoxide
co house
co objector
co officer
cod fish
cod swimmer
col neck
col pass
cole old king
colon stop
com commander
comb hairdresser
composer scorer
con Conservative
con against
con party
con politician
con study
con swindle
con trick
cooler prison
core decentralise
corn naval commander
cot bed
cot house
cow lower
cow neat
cr credit
cr crown
cr king
cs Civil Service
cs Czechoslovakia
cs hundreds
cs seas
ct Connecticut
ct carat
ct caught
ct cent/centime
ct court
ct small weight
ct weight
cu copper
cu see you
cue queue
cure priest (Fr.)
cutie pretty girl
cv autobiography
cy see why
d (old) penny
d Dee
d Democrat
d Deutsch
d Germany
d Schubert's works
d copper
d damn
d date
d daughter
d day
d dead/died
d deci
d degree
d delete
d delta
d deserted
d deuterium
d diameter
d diamond
d differential operator
d electrical flux
d five hundred
d four
d four thousand
d hundreds
d key +
d lot
d many +
d mark
d note +
d notice
d number
d strings (violin)
d violin string
d vitamin
da American lawyer
da District Attorney
da agreement - foreign (Russ.)
da dagger *
da lawman
da lawyer
da yes (Russ.)
dab expert
dad father
dad old man
dail Irish house
dam barrier
dam mother
dam restrain
dame lady
dan tribe
das articles - German
das the (Ger.)
dc Washington
dc Washington
dc current
dd cleric
dd days
dd divine
dd doctor
dd doctor of divinity
dd theologian
de from (Fr.)
de of (Fr.)
dean good man
dec Christmas period
dec last month
decanter Tantalus' prisoner
dee river
deed indeed
deed legal document
deep in the main
deep main
deep sea
deg degree
del of the (Ital.)
dela from the (Fr.)
dela of the (Fr.)
demi half
den retreat
den study
der articles - German
der the (Ger.)
derby horse race
des of the (Fr.)
det detective
di double
di five hundred and one
di princess
di two (double)
die articles - German
die the (Ger.)
dime 12.5 cents
dior designer
dis Hell
dis Pluto
dis underworld
disc circle
disc record
disc ring
dish pretty girl
dit named
dit reported
dit said (Fr.)
dit say (Fr.)
diy amateur's department
dk Denmark
dm mark
do (the) same
do act
do cheat
do cook
do ditto
do note
do party
do work
dodo double act
doh note
don fellow
don nobleman
don put on
don university teacher
down county
dr dead reckoning
dr doctor
dr dram
dr drawer
dr healer
drake bowler
dt alcoholic state
dt psychotic state
du from the (Fr.)
du of the (Fr.)
dutch wife
e Asian
e Edward
e Elizabeth
e England/English
e Spain
e boat
e bridge players
e direction
e east/eastern
e eight
e eight thousand
e energy +
e epsilon
e eta
e five
e five thousand
e key +
e layer
e logarithm base
e low grade
e note +
e orient
e oriental
e point
e quarter
e strings (violin)
e two hundred and fifty
e two hundred and fifty thousand
e universal set
e violin string
e vitamin
ea East Africa
ea each
ea river *
ea running water
ea water
ear listener
ear organ
ear spike *
earp lawman
eat Tanzania
ebor archbishop
ec London district
ec city
eccles Cakesville
ed Edward
ed editor
ed journalist
eden garden
eden old Prime Minister
eden paradise
edison inventor
edit censor
ee ease
eel fish
eel swimmer
eer always
eer ever
eer invariably
eg for example
eg for instance
egg bomb
egg cocktail
egg encourage
eight rowing boat
ein number one (Ger.)
el American railway
el American railway
el articles - Spanish
el measure
el printer's measure
el small measure
el the (Span.)
eld old age
eli priest
eli prophet
elia writer
ell four feet
ell length
ell measure
ely city
ely city
ely see
em measure
em printer's measure
em small measure
em small square
em them
en measure
en printer's measure
en small measure
eng England/English
ent otorhinolaryngology
entry record
eon age
eon time
ep record
er Cockney girl
er QE
er difficulty
er ever
er hesitation
er king
er monarch
er queen
er royal badge
era generation
erasmus old scholar
ere always
ere before
ergo so
eric gradually
erie lake
err blunder
err sin
err wander
erse Gaelic
es French art
es ease
esp sixth sense