1
Heu i: A Sc abble Playing Engine Using a P obabili y-Based Heu is ic
Alejand o González Rome o
a,∗
, René Alquéza Mancho
a
, A u o Ramí ez Flo es
b
,
F ancisco González Acuña
c,d
and Ian Ga cía Olmedo
e
a Ciències de la Compu ació, Uni e si a Poli ècnica de Ca alunya, Ba celona, Espanya
E-mails: [email p o ec ed], [email p o ec ed]
b Ciencias de la Compu ación, Cen o de In es igación en Ma emá icas A.C., Guanajua o, México
E-mail: [email p o ec ed]
c Ma emá icas Básicas, Cen o de In es igación en Ma emá icas A.C., Guanajua o, Mexico
E-mail: [email p o ec ed]
d Ins i u o de Ma emá icas (Unidad Cue na aca), Uni e sidad Nacional Au ónoma de México,
Cue na aca, Mo elos, México
e Uni e sidad Nacional Au ónoma de México, Ciudad de México, México
E-mail: [email p o ec ed]
Abs ac . The game o Sc abble has been success ully ackled by wo engines: Quackle and Ma en. They a ain he s a e-o - he-
a in Compu e Sc abble. These engines use simula ion echniques and p ecalcula ed alues o achie e supe human play. This
pape p esen s a Sc abble-playing engine, Heu i, which achie es wo ld championship s anda ds when playing in Spanish, and a
e y high le el when playing in English and F ench, wi hou using b u e- o ce app oaches.One ad an age o Heu i is ha i s
calcula ions, as opposed o many in Quackle, a e no p ecompu ed and ins ead pe o med o e e y u n. Spanish ma ch esul s o
di e en e sions o Heu i agains humans and wo e sions o Quackle a e p esen ed in his pape . Resul s agains Quackle, in
English and F ench, a e also gi en. The main goal o his pape is o en ich Compu e Sc abble wi h he desc ip ion o a Sc abble
playing p og am, Heu i, which is di e en ly designed om ea lie ones such as Ma en and Quackle. An addi ional goal is o
gi e Sc abble playe s an excellen s a egy o ob aining championship le el in Spanish.
Keywo ds: Sc abble, bingo, heu is ic, bingo p obabili y, Heu i, lea e
1. INTRODUCTION
Sc abble is a wo d game in which, in i s ou namen e sion, wo playe s sco e poin s by placing iles, each ile
ep esen ing a le e o o m wo ds on a boa d, a he like in a c osswo d. Wo ld Championship Tou namen s a e
played in many languages, such as English, F ench, and Spanish, English being he mos common.
The AI game adi ion ocuses on games wi h pe ec in o ma ion, wi h Go, Chess, Checke s, Lines o Ac ion,
Re e se, Awa i, Hex, Renju, Amazons, Pen e, e c. domina ing he li e a u e.
A body o li e a u e also exis s on impe ec in o ma ion games (in o he wo ds, hose wi h p i a e in o ma ion),
such as Sc abble, Poke and B idge, o which his a icle con ibu es.
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I is deba able whe he a s a e-o - he-a Sc abble compu e p og am bea s he bes human. In Wiles (2018) Howa d
Wa ne s a es: “Nigel Richa ds, he op playe in he wo ld, seems o ha e compu e o a b ain. In many ways we
hink he could ac ually be supe io o Quackle he p og am.”
“Edi o ial no e: Acco ding o Fi eThi y Eigh ’s analysis o c oss. ables.com’s da a, Richa ds is echnically anked
highe han Quackle.”(Wiles, 2018).
The e is only one head- o-head encoun e be ween Nigel Richa ds and Quackle, i ook place in he To on o
In e na ional Sc abble Open 2011, Nigel won 473-449.
Spanish Sc abble playe s a e no as good as English Sc abble playe s, pe haps because English Sc abble has been
a ound since he mid 1970’s, and Spanish Sc abble since a ound 2000. Spanish Sc abble playe s can lea n a lo om
playing and wa ching games using s ong bo s. We hink ha humans can lea n a lo om a Sc abble Heu is ic bo
such as Heu i.
E en hough Heu i can play in any language, i a dic iona y is plugged in o i , i pe o ms be e wi h highly in lec ed
languages, such as Spanish and F ench, and is less e ec i e in languages whose e bs ha e ew in lec ions such as
English. The goal o his a icle is o en ich Compu e Sc abble wi h he desc ip ion o a Sc abble playing p og am,
Heu i, designed di e en ly om p e ious ones such as Ma en and Quackle. An addi ional goal o his pape is o gi e
Sc abble playe s an excellen s a egy o achie ing championship le el in Spanish.
Heu i is a Type B p og am. Acco ding o Shannon (1950), a Type B p og am is a selec i e app oach o sea ching in
minimax ees which only conside s a subse o plausible mo es, in con as o b u e- o ce, Type A p og ams.
Quackle, which has a s a e-o - he-a s a u e in Compu e Sc abble, has an engine ha is close o a Type A p og am.
Quackle plays hund eds o housands o sel -play games o p e-calcula e he alues o Supe lea es, which a e
e alua ions o all s ings o leng h 6 o less. On he o he hand, Heu i calcula es he alues o s ings, o leng h 6 o
less, he momen a playe has o make a mo e (wi h he ac ual boa d).
Heu i has achie ed ex emely good esul s in Spanish, agains humans and Quackle, and easonable esul s in English
o F ench, o a Type B p og am, agains Quackle.
The a icle is o ganized in o se e al sec ions. Sec ion 2 men ions he mo e gene a o me hods, desc ibes he s a e-o -
he-a in Compu e Sc abble, and includes some esul s o Heu i ma ches in Spanish agains op-le el Sc abble
playe s and Quackle. Sec ion 3 e e s o he i s Spanish Sc abble p og am played wi h he o icial Fede ación
In e nacional de Sc abble en Español (FISE) ules. Sec ion 4 deals wi h he heu is ics used in he i s , second and
hi d e sions o Heu i and includes e alua ion o mulas as well as he ishing echnique. Ma ch esul s a e p o ided
in Sec ion 5, oge he wi h a compa ison o esul s using di e en languages. Las ly, Sec ion 6 con ains conclusions
and sugges ions o u u e esea ch.
2. SCRABBLE ENGINES
In he 1980’s when compu e s had less memo y, Sc abble engines used a Di ec Acyclic Wo d G aph, DAWG
(Appel and Jacobson,1988), an e icien da a s uc u e o s o ing he Sc abble lexicon, which was bo h ingenious
and memo y sa ing. Today’s compu e memo ies, howe e , make i possible o s o e Sc abble lexicons in se e al
iles wi h a mo e memo y han DAWGs. The Anag am Me hod (González e al., 2018) uses his ype o s o age o
inc ease he speed o mo e gene a ion. Wi h he Anag am Me hod, mo e gene a ion becomes as e han adi ional
mo e gene a o me hods using DAWG o GADDAG (Di ec ed Acyclic G aph p e ixed by i s own in e se) see
Go don (1994).
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Once a mo e gene a o exis s, he implemen a ion o basic engines based on he mo e ha maximizes he sco e in
each u n is he nex logical s ep. Shapi o (1979) and S ua (1982) a e examples o o e unne s in he de elopmen
o Sc abble engines. Simula ion echniques we e used o o e come he ob ious sho comings o his g eedy
app oach. These echniques conside ed he possible eplies o a playe ’s candida e mo e, he eplies o hose
eplies, and so on o many plies. This me hod, known as simula ion, elies on Mon e Ca lo sampling (B ownlee
Jason, 2019) o gene a e he opponen ’s ack, in si ua ions wi h unce ain y, i s heo e ical basis being he Minimax
echnique (Russell and No ig, 2009) o Game Theo y.
The Ma en p og am (Sheppa d, 2002) de eloped by B ian Sheppa d is one o he e e ences o his pa adigm, as
bo ne ou by i s excellen esul s agains op-le el playe s. Some yea s la e , Ka z-B own and O’Laughlin (2007)
implemen ed Quackle, a powe ul a i icial in elligence Sc abble playe , which also exploi s he simula ion
echnique.
Ma en cons uc s a da a se , gi ing a nume ical alue o se s o iles, ypically o leng h 1,2,3 o 4. The e alua ion
o a lea e is ob ained, oughly, by me ging his wi h o he numbe s in ol ing heu is ics, such as he quo ien
ob ained by di iding he numbe o owels by he numbe o consonan s (Sheppa d, 2002).
Quackle does he same hing concep ually, he di e ence being ha i now conside s nume ical alues o all
s ings o leng h less han 7. These alues a e calcula ed by playing hund eds o housands o Quackle s. Quackle
games (Ka z-B own and O’Laughlin, 2007).
In addi ion o he lea e e alua ions men ioned abo e, he s ong ea u es o Ma en and Quackle include hei
simula ion p ocedu es and, in he case o Quackle, he powe o hei Supe lea es.
A weakness o Ma en and Quackle is he ac ha hei calcula ions a e p ecompu ed, and do no ake in o accoun
he ac ual boa d, as opposed o Heu i’s calcula ions, which a e pe o med e e y u n, using he ac ual boa d.
The au ho s ha e de eloped Heu i (González e al., 2012; Ramí ez e al., 2009; González e al., 2008), an engine
ha uses he Anag am Me hod (González e al., 2018) o gene a e all he legal mo es. Unlike Ma en and Quackle,
o da e, Heu i has no used simula ion o decide which mo e o make. Ins ead, Heu i uses a p obabili y-based
heu is ic o choose he mo e o be played. Se e al ma ches agains op-le el Sc abble playe s ha e con i med
Heu i’s high playing s eng h.
Heu i’s mos impo an ma ch ic o y agains humans, was agains he 2006 and 2008 Wo ld Sc abble Champion
En ic He nandez. Heu i de ea ed he champion 6-0!
In Spanish, Heu i played a 2000-game ma ch agains Quackle using he Speedy Playe , winning 61% o he games.
The Speedy Playe e alua es po en ial mo es wi hou o wa d looking, and Quackle is weake in Spanish since
he e a e no p e-compu ed lea e alues (es ima ed alues o he le e s le on he playe ’s ack) o Spanish. Lea e
alues a e a key ea u e o Quackle’s playing s eng h.
Heu i also played a 1014-game ma ch agains Quackle using he 20-Second Championship Playe bo . This is a
much s onge bo han he Speedy Playe one, since i uses a 20-second simula ion o e e y mo e. Heu i won
53.35% o he games.
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3. THE BEGINNING
3.1. A Spanish Sc abble P og am
A e a g ea deal o wo k, he eno mous ask o building he lexicon was comple ed. We hen modi ied Sc abble ,
a ee English Sc abble sou ce p og am buil by Ami abh in 1999. The double le e s CH, LL, RR we e added as
well as he le e Ñ. The dis ibu ion was also changed and inally some modi ica ions had o be made o adap o
he ules o Sc abble ou namen s in Spanish, in ol ing he abili y o a playe o choose whe he o change a
ce ain numbe o iles o o pass.
E en ually, we had he i s Spanish Sc abble p og am eady o play wi h he o icial Spanish Sc abble ou namen
ules (FISE ules). Le us call his p og am Sc abble II.
To check ou Spanish lexicon and he unc ionali y o he Sc abble II p og am, a ma ch was played be ween
Sc abble II and an expe human Sc abble playe anked 28 h in he Spanish Sc abble league and 90 h in Wo ld
Spanish Sc abble. The ma ch ollowed he o ma o a baseball Wo ld Se ies ma ch. In o he wo ds, he i s playe
o win ou games wins he ma ch. The human expe won 4-2.
Al hough a i s sigh one migh hink ha a compu e has an eno mous ad an age o e a human because i knows
all he pe missible wo ds, i u ns ou ha his does no su ice o bea a Sc abble expe . This is because Sc abble
II adop s a nai e, g eedy s a egy. I always plays a mo e ha will sco e he highes numbe o poin s a ha
momen . This is an op imal s a egy o duplica e Sc abble, whe e he aim o he game is o make a mo e ha
ob ains he mos poin s e e y u n. Howe e , his is no a good s a egy o no mal Sc abble, since i igno es how
his could a ec he possibili y o making be e mo es in u u e plays. Sc abble II o en ends up wi h poo quali y
acks such as GGLHQPM and since he e is no exchange s a egy, hese acks will end o p e ail, gi ing he
human playe an eno mous ad an age. Fo ins ance, in Sc abble, a good playe usually ies o play a bingo (all
se en iles in one u n), since his gi es 50 bonus poin s. I is he e o e ad isable o keep he blank iles, which can
be played ep esen ing a single chosen pe missible le e . Since Sc abble II always plays a mo e wi h he highes
possible sco e, i ends o ge id o blank iles wi hou necessa ily making a bingo.
Since all hese ea u es we e obse ed, we decided o design an imp o ed e sion o Sc abble II wi h an in e ence
s a egy engine. This engine uses a heu is ic unc ion ha a emp s o make high sco ing mo es while balancing he
ack.
4. MOVE EVALUATOR
4.1. In oduc ion o Heu i
The i s s ep owa ds building a Sc abble engine, wi h an in e ence s a egy ha could balance he ack in play,
was a C unc ion ha ecei ed a 7-le e s ing and calcula ed he p obabili y o making a bingo, o all 128 possible
exchanges o he gi en s ing. To his end, we buil a ile, he sep e s, con aining all he 7-le e s ings ha p oduce
a bingo. An example o how o calcula e his p obabili y is gi en in sec ion 4.2.
An essen ial pa o e alua ing a pa icula mo e in ol es assessing wha will be le on he ack a e he mo e has
been made ( he ack esidue o lea e). Since he e is a 50-poin bonus when you use all you iles, a na u al way o
assess a mo e is o add i s sco e o he expec ed alue ha he lea e will be able o p oduce a bingo in he ollowing
mo e.
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Se e al ideas abou he heu is ics used by humans when playing Sc abble, such as he use o anag ams, led o he
cons uc ion o he compu e lexicon and he Heu i engine.
4.2. P obabili y and heu is ic o Heu i’s i s engine
An impo an pa o a p og am ha plays Sc abble in Spanish (o any o he language) is deciding wha o lea e on
he ack. In González e al. (2008), we p o ide a nume ical e alua ion as ollows:
= j + p * b − d (1)
whe e j is he numbe o poin s made by he mo e, in which iles a e played (i is assumed he e ha he bag has a
leas iles); j = 0 i he iles a e exchanged a he han played on he boa d; p is he p obabili y o ob aining a 7-
le e bingo i iles a e d awn a andom om a se ha is he union o he bag and he opponen ’s ack (which we
call he “augmen ed bag”); b, he expec ed alue o a bingo, is 77 (an a e age ob ained om 1000 games) o a he :
b = 50 + 2.5( + 1.92 ) (2)
whe e is he numbe o iles d awn om he bag and is he o al numbe o poin s in he lea e; d is a nonnega i e
numbe , which is ze o i he mo e is no weak om a de ensi e poin o iew. I is 20, say, i one places a ile
be ween wo iple wo d sco e squa es, pe mi ing a possible nine- ime , 10 i one plays a owel nex o a p emium
squa e allowing a six- ime , and 5 i he play pe mi s a possible ou - ime .
The equali y o b was ob ained as ollows. Fi s , le b
=
50
+
k * e whe e e is he expec ed
o al o poin s
(wi hou mul iplie s o bonuses) o a bingo. The sum o he alues o he 100 iles is 192, meaning ha he a e age
alue o a ile is 1.92 and e = + 1.92 . Fo example, i he lea e is {ZADO},
wi h
=
14, and we d aw
=
3
iles om he bag, we would expec hese iles o add, on a e age,
* 1.92 = 3 * 1.92. Say hese iles a e {VES}
(adding 6 poin s). Toge he wi h {ZADO} hese iles
o m VEZADOS, gi ing e
=
20 (app oxima ely
+
1.92 ).
Bu mos bingos include p emium squa es
( e y o en, in addi ion o he 50-poin bonus, he wo d sco e is
mul iplied by 2 and some imes 3, 4 o 9). I is he e o e easonable o mul iply e by a cons an k and we ook k
=
2.5 because i o en yields a b alue close o he expe imen al bingo a e age o 77.
Fo example, he equen case whe e
=
3,
= 4 yields b = 50 + 2.5(3 + 1.92 *
4) = 76.7.
I is be e o explain he calcula ion o p using an example:
Wha is he p obabili y ps o ob aining he “sep e ” s ={AAAAÑRR} ( om which one can o m he 7-le e bingo
ARAÑARA) i one has a lea e {AAAÑ}, exchanges {HQV}, he augmen ed bag is “ o al bag – {AAAÑHQV}”
and d aws h ee iles om i ?
Answe : I {AAAÑ} we e no con ained in {AAAAÑRR} p would be 0. Howe e {AAAÑ} is con ained in
{AAAAÑRR} so we conside he di e en se {AAAAÑRR}-{AAAÑ}={ARR}={AR2}
and
he
augmen ed
bag:
{AAAAAAAAARRRBBCCCC...XYZ}={A9R3B2C4...XYZ}.
One
hen compu es ps= C(9, 1) * C(3, 2) / C(93, 3)
(93 is he numbe o iles in he augmen ed bag and he 3 in C(93, 3) is he numbe o iles d awn om he bag)
whe e C(m, n) is he binomial coe icien :
C (m, n) = m (m − 1) (m − 2) . . . (m − n + 1) / n!
(3)

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The nume a o has n ac o s and C(m,n) is he numbe o n-elemen subse s o a se consis ing o m elemen s.
No e ha he denomina o C(93,3) does no depend on he sep e s.
The p obabili y p o ob aining a 7-le e bingo i one has he lea e {AAAÑ} and he augmen ed bag " o al bag -
{AAAÑHQV}" is he e o e he sum o all ps when s uns o e all sep e s.
4.3. Heu i’s second engine
Al hough Heu i’s i s engine has he essen ial pa o he algo i hm used by Heu i o e alua e a mo e, i equi ed
ce ain imp o emen s. In addi ion o making he engine play be e , hese imp o emen s enabled he engine o s a
playing au oma ically.
In Ramí ez e al. (2009), i was decided o gi e a nume ical e alua ion o all po en ial mo es as ollows:
Le us ecall some essen ial ideas om he las sec ion and in oduce some new ea u es.
An impo an pa o a p og am ha plays Sc abble is he decision o wha o lea e on he ack. In a mo e iles
a e played o exchanged, and n - iles make up he lea e (excluding he endgame,
n
=
7).
The ollowing heu is ic unc ion is used o e alua e all po en ial mo es:
=
j
+
e − d (4)
whe e j is he numbe o poin s made by he mo e, in which iles a e played (i is assumed he e ha he bag
has a leas iles; j
=
0 i iles a e exchanged a he han played); e is he expec ed alue o a bingo, gi en a
lea e , i iles a e d awn andomly om he augmen ed bag, in o he wo ds,
he
union
o
he
bag
and
he
opponen ’s
ack;
d
is
a
nonnega i e
numbe , which
is
ze o
i
he
mo e is no weak om a de ensi e poin o iew.
In (4), d
=
0. Ou o all he po en ial mo es, one wi h maximal is chosen.
To explain ou es ima e o e, a sep e is de ined as a lexicog aphically o de ed s ing o se en cha ac
e s o
{A,B,...,Z,#} (whe e # is a blank) om which a 7-le e wo d can be cons uc ed. Fo example,
{AAAAÑRR}
(yielding ARAÑARA) and {ACEINR#} (yielding RECIBAN) a e sep e s bu {AEEQRY#} and {ADEILOS} a e
no .
The e a e 126,972 sep e s. The ollowing o mula was used o calcula e e:
e=∑
i=1
126,972
pi(50+k σi)
(5)
Le us call Heu i_2 he heu is ic in (5). Heu i_2 is he heu is ic used in he Heu i p og am p esen ed in Ramí ez e
al., (2009).
He e, pi is he p obabili y (which could be ze o) o ob aining he i- h sep e , gi en a lea e consis ing o 7 - iles,
i iles a e andomly d awn om he augmen ed bag; σi is he sum o he alues o he cha ac e s in he i- h sep e .
The exis ence o p emium squa es, hook wo ds, bingos o leng h g ea e han 7 and expe imen a ion ha e led us o
gi e 2.5 as he alue o he cons an k.
The calcula ion o pi was explained (wi h an example) in sec ion 4.2.
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4.4. Heu i’s hi d engine
Al hough he o mula (5) is a good es ima e o e when he boa d is open, i igno es he ac ha one may ha e a
sep e on he ack which canno be placed as a bingo on he boa d. This o en happens when he boa d is
closed. To accoun o his, one can w i e:
e=∑
i=1
126,972
δipi(50+k σi)
(6)
whe e δi, which depends on he boa d, is 1 i he i- h sep e can be played as a bingo on he boa d and 0
o he wise; ( he boa d is aken be o e he nex opponen ’s mo e). The calcula ion o δi, o all i, which is
independen o he ack, is easible because Heu i has a as mo e gene a o based on anag ams (González e al.,
2018).
A collec ion o 1000 Sc abble games was amassed o classi y he bingos played in hose games in o h ee
ca ego ies by leng h: 1) se en-le e bingos, 2) eigh -le e bingos, and 3) X-le e bingos, wi h X > 8. This
classi ica ion yielded he ollowing esul s:
The o al numbe o bingos in he collec ion o 1000 games was 6073 bingos, di ided as ollows: 2003 se en-le e
bingos, 3972 eigh -le e bingos and 98 bingos wi h mo e han 8 le e s. This ein o ced a p e ious hough : he
heu is ic unc ion should include he p obabili y o making 8-le e as well as 7-le e bingos.
Le us de ine an oc e as a lexicog aphically a anged
s ing o eigh cha ac e s o {A, B, ..., Z,#} (whe e
# is a blank)
om which an 8-le e wo d can be cons uc ed.
A educed oc e is a lexicog aphically o de ed s ing o se en cha ac e s o {A, B, ..., Z, #}, which o ms
an oc e when
adding a sui able new cha ac e . Le a pu e sep e be a sep e ha is no a educed oc e .
The e a e 126,972 sep e s, 5,830 pu e sep e s and 282,214 educed oc e s. No e ha he se o sep e s con ains all
he pu e sep e s and 121,142 educed oc e s ha a e also sep e s. We wish o p o ide an es ima e o he
p obabili y o ob aining a 7-o 8-le e bingo, gi en a lea e and a boa d. The pu e sep e s and he educed oc e s
yield a o al o 288,044.
Then he calcula ion o e is gi en in González e al. (2012) by he o mula:
e=∑
i=1
288,044
δipi(50+k σi)
(7)
Le Heu i_3 be he heu is ic in (7); pi, σi, k and δi we e explained ea lie .
4.5. Fishing
Fishing plays a e impo an because hey seek high sco ing mo es in subsequen u ns. B ian Sheppa d, he
de elope o Ma en, w i es he ollowing in Sheppa d (2002): “Ma en’s lack o a mo e gene a o o ishing may
be i s bigges weakness. All o he mo es ha Ma en o e looked in i s ma ch agains Adam Logan we e ish.”
Heu i ishes ex emely well, which is a key ea u e o a good Sc abble engine.
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In ui i ely, a ish is a mo e ha seeks a bingo in he nex u n. Mo e p ecisely, using Heu i’s p obab
ili y heu is ic,
=
j +e -
d, le us de ine a ish as a mo e whe e e > j (i s expec ed alue o a bingo is g ea e han he poin s
made by he mo e). The mos common example o a ish is an exchange wi h e > 0. Almos all exchanges a e
ishing mo es. Howe e , in he p e-endgame (when he e a e only a ew iles in he bag), a playe may change a Q
ile o a oid being le wi h i a he end. I is likely ha e
=
0 since no mo e bingos can be placed on he boa d, o
i is impossible o cons uc a bingo wi h he playe ’s ack lea e and he iles le inside he bag (González e al.,
2012).
The ollowing is an example o a possible line o play, using Heu i, which shows and in ol es ishing mo es.
Figu e 1 shows he ishing mo es; he playe ’s ack is e uuqh. The i s six mo es a e ishing mo es since e > j.
All mo es excep mo e 7 a e ishing mo es since e > j o all o hem. In 7 hu e,
j
=
24 and e
=
9.515 so j >
e. (In igu e 1, p ep esen s he alue o e).
Fig. 1. Fishing pa 1.
A e playing he bes ishing mo e, (7G he) a le e “e” is d awn, om he bag, and he boa d posi ion is shown in
igu e 2.
Fig. 2. Fishing pa 2.
The nex playe hen exchanges iles. The playe in u n has he ollowing ack: e uuqe and he bes mo e is
shown in igu e 3 (8A u quees which gi es 106 poin s). Ano he equally good mo e is 8A uquees.
9
Fig. 3. Fishing pa 3.
5. MATCH RESULTS
5.1. Ma ch Resul s o Heu i’s Fi s Engine
Due o he un inished s a e o he engine, we could only pe o m a ew games o es ou i s heu is ic. The i s
ma ch Heu i played was a ma ch agains an engine ha always played he highes mo e: Heu i won 4-0. Ma ches
agains human beings we e pe o med on he In e ne , which ga e hem an ad an age since hey could consul
wo ds be o e placing hem on he boa d and also had a lo o ime o play. In i s i s ma ch agains a human being,
Heu i won 4-2 agains one o he bes playe s in Colombia, Fe nando Man íquez, wi h an ELO o 2110 ( he highes
ELO in he FISE lis was 2191), anked No.20 in he FISE lis o Sc abble playe s. Ano he impo an ic o y Heu i
sco ed was agains Benjamín Olaizola, om Venezuela, he eigning (2007) wo ld champion who had also won he
wo ld championship in 2001. Wi h an ELO o 2169, he anked second in he in e na ional lis . I was a e y close
game: (Heu i 471 Benjamín Olaizola 465). E en ually, Heu i los he ma ch o Olaizola 2-4. A e Heu i played 17
games agains se e al op- anked playe s, he esul s we e as ollows: Heu i won 10 and los 7, wi h an a e age o
509 poin s pe game.
5.2. Ma ch Resul s o Heu i’s Second Engine
A e playing 40 games agains se e al op- anked playe s, Heu i achie ed he ollowing esul s using he heu is ic
Heu i_2 desc ibed in sec ion 4.3: 27 wins and 13 losses, wi h an a e age game sco e o 511 poin s. Almos all he
ma ches ollowed he o ma o a baseball wo ld se ies: he i s opponen o sco e ou ic o ies won he ma ch.
The mos impo an ma ch ic o y o Heu i was agains 2008 Wo ld Champion En ic He nández, om Spain, also
Wo ld Champion in 2006. This was a longe ma ch, consis ing o a maximum o ele en games o decide he
winne . Heu i de ea ed he Wo ld Champion 6-0! An excellen esul . Heu i also won 4-1 a ma ch agains Ai an
Pé ez om Venezuela, one o he bes playe s in he wo ld, and 2007 and 2008 Runne -up. Pé ez has won wo
Wo ld Championships (in 2013 and 2015). Heu i also de ea ed Aglaia Cons an in, hen he bes playe o
Colombia, wice Na ional Champion o Colombia (2006 and 2009), 4-1. Heu i won ou ma ches and los one: he
emainde o he games we e incomple e ma ches agains o me wo ld champions and na ional champions.