scacchi e matematica mathesis -...
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![Page 1: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/1.jpg)
Scacchi e MatematicaMathesis
13/01/2010
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Origini
• India ≈ VI secolo - Chaturanga• Quadruplice Armata: Fanti, Cavalieri, Elefanti, Carri• ⇒ Persia⇒ Mondo Islamico⇒ Europa
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• ≈ 1000 Europa (Spagna, Sicilia, ...)• Forma attuale: ≈ 1400
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Notazione Posizionale
• India ≈ VI secolo :NotazionePosizionale ( eZero)
• ⇒ Persia ⇒Mondo Islamico⇒ Europa Al-
Khwarizmi - IXsecolo
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Primi Campioni
• I Torneo: Londra 1851 (Adolf Anderssen)Laurea in Matematica all’Università di Breslau; insegnamatematica Friedrichs Gymnasium dal 1847 dal 1879.
• I Campione del mondo: Wilhelm Steinitz; 1886–1894
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• Emanuel Lasker - Scacchista e Matematico
Campione del Mondo dal 1894 al 1921.Studia matematica e filosofia a Berlino e Heidelberg.Dottorato in matematica a Erlangen (1900–1902), su consigliodi D. Hilbert
• Lasker ha ottenuto importanti risultati in Algebracommutativa:E. Lasker –E. Noether, Teoremi di Decomposizione primaria diIdeali
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Altri Scacchisti-Matematici
• Machgielis (Max) Euwe (1901-1981).
Campione del Mondo 1935-1937• J. Nunn (1955-), uno dei più forti grandi maestri degli ultimianni. PhD in Topologia Algebrica ad Oxford,
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Scacchi
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• Insieme di Regole Semplici;• Universo finito;• ⇒ Enorme Complessità
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La Ricompensa
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• 1 chicco di riso sulla prima casa;• 2 chicchi sulla seconda casa;• 4 chicchi sulla terza casa;. . .
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1 + 21 + 22 + 23 + · · ·+ 263 = 264 − 1 =
= 18.446.744.073.709.551.615 ≈ 18 · 1018
1. Prima metà della scacchiera: 232 − 1 = 4, 294, 967, 295chicchi. Circa la produzione annuale dell’India.
2. Seconda metà della scacchiera:
264 − 232 = 232(232 − 1)
3. Solo nell’ultima casa:più di due miliardi di volte tutta la prima metà della scacchiera.
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Crescita esponenziale
Quante volte dobbiamo piegare un foglio di carta per ottenere unapila di carta alta quanto la distanza dalla terra al sole ?
• Un foglio di carta ha lo spessore di circa un decimo dimillimetro: 10−4 metri.
• Distanza terra–sole: ≈ 1.5 · 1011 metri.•
250 ≈ 1.1 · 1015
Risposta
51 volte
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Raddoppiare 11 volte ...
211 · 10−4m = 0.2048m ≈ 20 cm
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Complessità
C. Shannon (1950): Programming a Computer for Playing Chess
• Quante possibili partite ?• Quante possibili posizioni ?
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Stima del Numero di Partite; C. Shannon (1950)
Numero “tipico” di mosse: circa 30 (DeGroot, 1946: media su unconsistente numero di partite).⇒ circa 103 possibilità per turno Bianco-Nero.⇒ Considerando una lunghezza “tipica” di 40 mosse:
≈(103)40
= 10120
superiore al numero degli atomi dell’universo osservabile (tra4 · 1079 e 1081).
G.H. Hardy (1940)Hardy scrive (senza ulteriori spiegazioni) che “probabilmente, ilnumero di partite di scacchi è dell’ordine di grandezza”
≈ 101050
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Stima del Numero di Posizioni
C. Shannon; 1950Numero stimato di posizioni ’ammissibili’:
64!32! (8!)2 (2!)6
≈ 1043
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Numero di Posizioni dopo n (semi)mosse
• 1 semimossa: 20;• 2 semimosse: 400;• 3 semimosse: 5362;• 4 semimosse: 72.078• 5 semimosse: 822.518;• 6 semimosse: 9.417.681;• 7 semimosse: 96.400.068.
Fonte: The on-line Encyclopedia of integer sequences.Sequenza: A0832776 (fino a 9 semimosse).
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Torre
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Numero di casecontrollate:sempre 14
Forza relativa della Torre
1463
=29
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Regina
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Per le 28 case delbordo:14 + 7 = 21
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Per le 20 case del’secondo bordo’:14 + 9 = 23
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Per le 12 case del ’terzobordo’:14 + 11 = 25
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Per le 4 caselle delquadrato centrale:14 + 13 = 27
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Regina/Due Torri
Forza relativa della Regina
163
((28 · 21) + (20 · 23) + (12 · 25) + (4 · 27)
64
)=
=1336
29
+29
=1636
>1336
⇒ Due Torri sono più forti di una Regina
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EsercizioProblemaQual’e’ la probabilità che due regine poste a caso sulla scacchierasiano “indipendenti” (non si attacchino) ?Soluzione:Usiamo la formula della Probabilità Totale:
P(I ) =4∑
i=1
P(I |Ai )P(Ai )
dove Ai = la regina A si trova sull’ i-esimo bordo.• P(A1) = 28
64 , P(I |A1) = 4263 ;
• P(A2) = 2064 , P(I |A2) = 40
63 ;• P(A3) = 12
64 , P(I |A3) = 3863 ;
• P(A4) = 464 , P(I |A4) = 36
63 ;Quindi:
P(I ) =2864
4263
+2064
4063
+1264
3863
+464
3663
=25764032
≈ 0.639
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Problemi
1. Numero massimo di pezzi (di dato tipo) che si possonodisporre senza ’attaccarsi’ ? (Indipendenza)
2. Numero minimo di pezzi (di dato tipo) che ’attaccano’ o’occupano’ ogni casa ? (Dominazione)
3. Percorsi: il pezzo visita tutte le case una sola volta.
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Indipendenza - Regina su 8× 8
Su una scacchiera 8× 8 possono essere diposte 8 regine in modoche nessuna attacchi l’altra.
• Vi sono 92 modi distinti di disporle su una 8× 8.• 12 modi a meno di simmetrie (rotazioni e riflessioni).92 = (11 · 8) + 4
C.F. Gauss nel 1850 (in una lettera ad un amico) scrive di avertrovato 72 soluzioni, ma che ve ne possono essere ancora.Si veda anche: P. Campbell, Gauss and the eight queens problem,Historia Mathematica 4 (1997), 397–404.
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
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12 *
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n × n
Più in generale, qual’è il massimo numero di regine “indipendenti”(ovvero che non si attacchino a vicenda) che possono esseredisposte su una scacchiera n × n ?
W. Ahrens; 1901Su una scacchiera n × n, se n 6= 2, 3, possono essere disposte nRegine, in modo che nessuna attacchi l’altra.
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n × n: in quanti modi diversi ?
F (n): numero di soluzioni fondamentali.S(n): numero totale di soluzioni.
• F (8) = 12, S(8) = 92;• F (9) = 46, S(9) = 342;• F (10) = 92, S(10) = 724;• F (11) = 341, S(11) = 2680;• F (12) = 1784, S(12) = 14200.
![Page 39: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/39.jpg)
Dominazione con Regine
Su una scacchiera 8× 8sono necessarie, esufficienti, 5 Regine percoprire o occupare ognicasa
Vi sono 4860 modidistinti di disporre 5regine, su una 8× 8, inmodo che ’coprano’ lascacchiera (Yaglom eYaglom).
![Page 40: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/40.jpg)
Esempio
QZ0Z0Z0ZZ0Z0Z0Z00Z0Z0ZQZZ0L0Z0Z00Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0LZ0ZQZ0Z0
![Page 41: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/41.jpg)
Osservazione: quattro regine possono coprire 62 case!
0Z0Z0Z0ZZ0Z0Z0Z00Z0Z0ZQZZ0L0Z0Z00Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0LZ0ZQZ0Z0
![Page 42: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/42.jpg)
Esempio Diagonale
QZ0Z0Z0ZZ0Z0Z0Z00ZQZ0Z0ZZ0ZQZ0Z00Z0ZQZ0ZZ0Z0Z0Z00Z0Z0ZQZZ0Z0Z0Z0
![Page 43: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/43.jpg)
Altra configurazione “dominante” con 5 regine su una linea
0Z0L0Z0ZZ0Z0Z0Z00Z0Z0Z0ZZ0ZQZ0Z00Z0L0Z0ZZ0ZQZ0Z00Z0L0Z0ZZ0Z0Z0Z0
![Page 44: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/44.jpg)
Problema di Guarini (1512)
Scambiare di posto le due coppie di cavalli.
![Page 45: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/45.jpg)
Soluzione
La “traduzione” in termini di grafo rende la soluzione evidente.
![Page 46: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/46.jpg)
Grafi
Indipendenza
• Un sottoinsieme di vertici S di un grafo G si dice un insiemeindipendente in G se nessuno dei vertici di S è adiacente adalcun vertice di S .
• Si definisce numero di indipendenza
ι(G)
la massima cardinalità di un insieme indipendente di G.(Un insieme indipendente S di G si dice massimo se| S |= ι(G))
![Page 47: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/47.jpg)
Grafi
Dominazione
• Un sottoinsieme di vertici S di un grafo G si dice un insiemedominante in G se ogni vertice di G è in S oppure è adiacentead un vertice di S .
• Oss.: indipendente max. ⇒ dominante.• Si definisce numero di dominazione
δ(G)
la minima cardinalità di un insieme dominante di G.(Un insieme dominante S di G si dice minimo se | S |= δ(G))
![Page 48: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/48.jpg)
Grafo di Alfiere A4×4
L’insieme {1, 5, 9, 2, 4, 14} è un insieme indipendente di cardinalitàmassima di A4×4.Quindi ι(A4×4) = 6.L’insieme {7, 11, 10, 6} è un insieme dominante di A4×4.È di cardinalità minima: quindi δ(A4×4) = 4.
![Page 49: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/49.jpg)
Grafo di Regina Qn×n
Abbiamo visto che, per n 6= 2, 3,
ι(Qn×n) = n
mentre
ι(Q2×2) = 1 ι(Q3×3) = 2.
Problema aperto
δ(Qn×n) =?
![Page 50: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/50.jpg)
Osservazione: dominazione diagonale 6= dominazione
Il numero minimo di regine che domininano una scacchiera n × npuò variare se si richiedono condizioni aggiuntive, per esempio chestiano tutte sulla stessa diagonale.Ad esempio:
δ(Q11×11) = 5
madiagδ(Q11×11) = 7.
(mentre δ(Q8×8) = diagδ(Q8×8)).
![Page 51: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/51.jpg)
Indipendenza in Tn×n
• ι(Tn×n) ≤ n• ι(Tn×n) = n
RZ0Z0Z0ZZRZ0Z0Z00ZRZ0Z0ZZ0ZRZ0Z00Z0ZRZ0ZZ0Z0ZRZ00Z0Z0ZRZZ0Z0Z0ZR
![Page 52: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/52.jpg)
Numero di disposizioni indipendenti massime di torre
• Ognuna delle n torri deve trovarsi in righe e colonne diverse.• Abbiamo quindi
n!
disposizioni• Per la scacchiera usuale, con n = 8, abbiamo
8! = 40.320
possibilità.
![Page 53: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/53.jpg)
Dominazione in Tn×n
• δ(Tn×n) ≥ n• δ(Tn×n) = n
0Z0S0Z0ZZ0ZRZ0Z00Z0S0Z0ZZ0ZRZ0Z00Z0S0Z0ZZ0ZRZ0Z00Z0S0Z0ZZ0ZRZ0Z0
![Page 54: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/54.jpg)
Numero di configurazioni dominanti minime di torre
• Deve esserci una torre in ogni riga o una torre in ogni colonna.• Una torre in ogni riga: nn possibilità.• Una torre in ogni colonna: nn possibilità.• Per la scacchiera usuale n × n, abbiamo
nn + nn − n!
possibilità.• Per la scacchiera usuale, con n = 8, abbiamo
2 · 88 − 8! = 33.514.312
possibilità.
![Page 55: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/55.jpg)
Indipendenza in A8×8
ι(A8×8) ≤ 14
![Page 56: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/56.jpg)
Indipendenza in A8×8
ι(A8×8) = 14
![Page 57: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/57.jpg)
Indipendenza in An×n
•ι(An×n) = 2n − 2
• Si può dimostrare che in ogni configurazione indipendentemassima gli alfieri devono stare tutti sul bordo.
Il numero diconfigurazioniindipendenti massime inAn×n è 2n.
![Page 58: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/58.jpg)
Dominazione in A8×8
•• Bianco e nero: separatamente.
•Alfiere ⇒ Torre
• δ(A8×8) ≥ 4 + 4 = 8
![Page 59: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/59.jpg)
Dominazione in A8×8
δ(A8×8) = 8
0Z0A0Z0ZZ0ZBZ0Z00Z0A0Z0ZZ0ZBZ0Z00Z0A0Z0ZZ0ZBZ0Z00Z0A0Z0ZZ0ZBZ0Z0
![Page 60: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/60.jpg)
Dominazione in An×n
•δ(An×n) = n
• Sono note formule per il numero di configurazioni dominantiminime di Alfiere su n × n. Però sono complicate (si vedaYaglom e Yaglom, pagg. 83–88):
•
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Indipendenza in C8×8
ι(C8×8) ≥ 32
nZnZnZnZZnZnZnZnnZnZnZnZZnZnZnZnnZnZnZnZZnZnZnZnnZnZnZnZZnZnZnZn
![Page 62: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/62.jpg)
Indipendenza in C8×8
• In ogni rettangolo,un cavallocontrollaesattamenteun’altra casa delrettangolo.
• In ogni rettangoloci possono essereal piu’ 4 cavalli“indipendenti”
• Quindi
ι(C8×8) = 32
![Page 63: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/63.jpg)
Indipendenza in Cn×n
• Per n 6= 2 (ι(C2×2) = 4) si ha:• se n è pari
ι(Cn×n) =n2
2• se n è dispari
ι(Cn×n) =n2 + 1
2
![Page 64: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/64.jpg)
Dominazione in C8×8
0Z0Z0Z0ZZ0Z0ZnZ00mnZnm0ZZ0m0Z0Z00Z0Z0m0ZZ0mnZnm00ZnZ0Z0ZZ0Z0Z0Z0
12 cavalli sono sufficienti per dominare una scacchiera
![Page 65: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/65.jpg)
12 Cavalli
0Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0ZZ0Z0Z0Z0
• 12 cavalli sono anche necessari per 8× 8.Infatti, un cavallo può coprire al più una ×-casa
• ⇒ δ(C8×8) = 12.
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n = 11 (1971 !)
Problema aperto: determinare δ(Cn×n) per ogni n.
![Page 67: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/67.jpg)
Percorsi Cavallo: 8× 8
Percorso chiuso (ciclo) Percorso aperto
![Page 68: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/68.jpg)
C8×8
Sono cicli (risp. cammini ) hamiltoniani in C8×8.
![Page 69: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/69.jpg)
Eulero (1759)
Consiste in due cammini aperti, nelle metà inferiore e superioredella scacchiera, che si congiungono (ha simmetria centrale).
Esistenza ciclo di cavallo su 8× 8 ⇒ solo due configurazioniindipendenti massime di Cavallo in 8× 8.
![Page 70: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/70.jpg)
In generale
• Ci sono percorsi (chiusi) di cavallo su scacchiere m × n ?• Non sempre: ad esempio 3× 3.La casa centrale non è ’connessa’ alle altre.
![Page 71: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/71.jpg)
Divagazione: bianco e nero.E’ possibile tassellare con tessere del domino ?
![Page 72: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/72.jpg)
Risposta: No
![Page 73: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/73.jpg)
Bianco e nero
Il cavallo alterna case bianche e nere⇒ Non esistono percorsi chiusi di cavallo su scacchere m × n se me n sono entrambi dispari, perchè il numero di case bianche e nere è
diverso !
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5× 5
Ma esistono percorsi aperti su 5× 5.
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3× 6
• Rimuovendo i vertici A e B da C3×6 si ottengono trecomponenti connesse.
• ⇒ non può esserci un ciclo hamiltoniano in C3×6 (se da unacollana si rimuovono k grani, si formano al più k pezzi).
![Page 76: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/76.jpg)
3× 8
• Gli 8 vertici neri sono quelli di grado 2 in C3×8
• Un ciclo hamiltoniano in C3×8 deve necessariamente passareper i lati indicati.
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3× 8
• Tenendo conto che non ci possono essere sottocicli, e dellasimmetria, ci si riconduce al cammino parziale indicato.
• ⇒ non può esserci un ciclo hamiltoniano in C3×8
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4× n
Non esistono percorsi chiusi di cavallo su scacchiere 4× n.
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Dimostrazione!
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Dimostrazione!
• A → C ; C → A,D• B → D ; D → B,C• Se esistesse un ciclo hamiltoniano, lo potremmo ’aprire’ in uncammino hamiltoniano che inizia con A e termina con C.
• Ma per andare in B o D deve partire da C, e poi ritornarvi ! Siottiene una contraddizione con #A = #C .
• ⇒ non esiste un ciclo di cavallo su 4× n.
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3× 4
Ma esistono percorsi aperti su 4× 3 ( o 3× 4)
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Dimostrazione alternativa: bianco e nero.
Il cavallo alterna trabianco e nero.
Solo da bianco si arrivain nero.
Se esistesse un ciclo hamiltoniano, le due colorazioni dovrebberocoincidere!
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Terza Dimostrazione !
Non può esistere un ciclo di cavallo su 4× n perchè ci sono più didue configurazioni indipendenti massime (quella raffigurata, oltre
alle due ’monocromatiche’).
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Cicli di cavallo: in generale
Schwenk (1991)In una scacchiera m × n (m ≤ n) esiste sempre un ciclo di cavallo,con le sole seguenti eccezioni:
• m e n entrambi dispari;• m = 1, 2 oppure 4;• m = 3 e n = 4, 6 oppure 8.
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Percorsi aperti di cavallo: in generale
Cull-Curtnis (1978); Chia-Ong (2005)In una scacchiera m × n (m ≤ n) esiste sempre un percorso apertodi cavallo, con le sole seguenti eccezioni:
• m = 1 oppure 2;• m = 3 e n = 3, 5 oppure 6.• m = 4 = n.
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Distanza di Cavallo (su scacchiera illimitata)
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1 mossa
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2 mosse
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3 mosse
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4 mosse
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5 mosse
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6 mosse
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7 mosse
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8 mosse
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9 mosse
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10 mosse
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Palle di Cavallo (su scacchiera infinita)
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raggio 1
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raggio 2
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raggio 3
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raggio 4
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raggio 5
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raggio 6
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raggio 7
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raggio 8
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raggio 9
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raggio 10
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d(n, m)
N. ElkiesLa distanza di cavallo d(m, n) da (0, 0) a (m, n) è
d(m, n) =
⌈max
{| m |2
,| n |2,| m | + | n |
3
}⌉se (m, n) 6= (±1, 0), (0,±1), (±2,±2).Mentre
d(m, n) =
⌈max
{| m |2
,| n |2,| m | + | n |
3
}⌉+ 2
se (m, n) = (±1, 0), (0,±1), (±2,±2).
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Distanza di Cavallo su 8× 8
La casa marcata con ∗ è a distanza 4 per la particolarità dell’angolo.
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Il nero vince (con o senza tratto)
La casa a2 ha distanza 3 e la casa a1 ha distanza 4 !
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Il Bianco muove e vince.
Solo la reginanera si puòmuovere(a1− a2) e ilpedone bianco(se il re biancosi sposta i pezzineri si’sciolgono’).
8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 0opspZ0O1 lnjbJ0Z0
a b c d e f g h
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Strategia vincente.
L’unicapossibilità divittoria è darematto (dicavallo) in b3,dopo avereliminato ipedoni in b5 ec4. Occorreche però laregina nera siain a1 !
8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 0opspZ0O1 lnjbJ0Z0
a b c d e f g h
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Errore !
In questo modoil bianco puòsolo pattare.
8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0O3 aropZ0Z02 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h�
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Parità
h8 si trova adistanza dicavallo disparida b3.
8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 0opspZ0O1 lnjbJ0Z0
a b c d e f g h
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Soluzione
Si devemuovere ilpedone solo diun passo.
8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0ZP2 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0ZP2 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0ZP2 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0O3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
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a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0ZP4 0opZ0Z0Z3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0O5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
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a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0ZP6 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0ZP6 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0M7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0M7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0ZNZ06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0ZNZ06 0Z0Z0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0M0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0M0Z0Z5 ZpZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZNZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 ZNZ0Z0Z04 0opZ0Z0Z3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0M0Z0Z5 Z0Z0Z0Z04 0opZ0Z0Z3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
![Page 135: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/135.jpg)
8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0M0Z0Z5 Z0Z0Z0Z04 0opZ0Z0Z3 aropZ0Z02 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 Z0Z0Z0Z04 0oNZ0Z0Z3 aropZ0Z02 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
![Page 137: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/137.jpg)
8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 Z0Z0Z0Z04 0oNZ0Z0Z3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
![Page 138: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/138.jpg)
8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 M0Z0Z0Z04 0o0Z0Z0Z3 aropZ0Z02 qopspZ0Z1 ZnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 M0Z0Z0Z04 0o0Z0Z0Z3 aropZ0Z02 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
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8 0Z0Z0Z0Z7 Z0Z0Z0Z06 0Z0Z0Z0Z5 Z0Z0Z0Z04 0o0Z0Z0Z3 aNopZ0Z02 0opspZ0Z1 lnjbJ0Z0
a b c d e f g h
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Colora i Pezzi
0Z0Z0ZKLZ0Z0Z0Z00Z0Z0ZRJZ0Z0Z0Z00Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0ZZ0Z0Z0Z0
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Soluzione
0Z0Z0ZkLZ0Z0Z0Z00Z0Z0ZRJZ0Z0Z0Z00Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0ZZ0Z0Z0Z0
![Page 143: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/143.jpg)
Mossa precedente
8 0Z0Z0Zkm7 Z0Z0Z0O06 0Z0Z0ZRJ5 Z0Z0Z0Z04 0Z0Z0Z0Z3 Z0Z0Z0Z02 0Z0Z0Z0Z1 Z0Z0Z0Z0
a b c d e f g h
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Posizione dopo la quarta mossa del nero. Come è andato ilgioco ?
8 rmblka0s7 opo0Zpop6 0Z0Z0Z0Z5 Z0Z0Z0Z04 0Z0Z0Z0Z3 Z0Z0Z0Z02 POPOPOPO1 SNAQJBZR
a b c d e f g h
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1 Nf3
8 rmblkans7 opopopop6 0Z0Z0Z0Z5 Z0Z0Z0Z04 0Z0Z0Z0Z3 Z0Z0ZNZ02 POPOPOPO1 SNAQJBZR
a b c d e f g h
![Page 146: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/146.jpg)
1 Nf3 e5.
8 rmblkans7 opopZpop6 0Z0Z0Z0Z5 Z0Z0o0Z04 0Z0Z0Z0Z3 Z0Z0ZNZ02 POPOPOPO1 SNAQJBZR
a b c d e f g h
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1 Nf3 e5. 2 NXe5
8 rmblkans7 opopZpop6 0Z0Z0Z0Z5 Z0Z0M0Z04 0Z0Z0Z0Z3 Z0Z0Z0Z02 POPOPOPO1 SNAQJBZR
a b c d e f g h
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1 Nf3 e5 2 NXe5 Ne7
8 rmblka0s7 opopmpop6 0Z0Z0Z0Z5 Z0Z0M0Z04 0Z0Z0Z0Z3 Z0Z0Z0Z02 POPOPOPO1 SNAQJBZR
a b c d e f g h
![Page 149: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/149.jpg)
1 Nf3 e5 2 NXe5 Ne7 3 NXd7
8 rmblka0s7 opoNmpop6 0Z0Z0Z0Z5 Z0Z0Z0Z04 0Z0Z0Z0Z3 Z0Z0Z0Z02 POPOPOPO1 SNAQJBZR
a b c d e f g h
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1 Nf3 e5 2 NXe5 Ne7 3 NXd7 Nec6
8 rmblka0s7 opoNZpop6 0ZnZ0Z0Z5 Z0Z0Z0Z04 0Z0Z0Z0Z3 Z0Z0Z0Z02 POPOPOPO1 SNAQJBZR
a b c d e f g h
![Page 151: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/151.jpg)
1 Nf3 e5 2 NXe5 Ne7 3 NXd7 Nec6 4 NXb8
8 rMblka0s7 opo0Zpop6 0ZnZ0Z0Z5 Z0Z0Z0Z04 0Z0Z0Z0Z3 Z0Z0Z0Z02 POPOPOPO1 SNAQJBZR
a b c d e f g h
![Page 152: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/152.jpg)
1 Nf3 e5 2 NXe5 Ne7 3 NXd7 Nec6 4 NXb8 NXb8
8 rmblka0s7 opo0Zpop6 0Z0Z0Z0Z5 Z0Z0Z0Z04 0Z0Z0Z0Z3 Z0Z0Z0Z02 POPOPOPO1 SNAQJBZR
a b c d e f g h
![Page 153: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/153.jpg)
Posizione dopo la quarta mossa del nero. Come è andato ilgioco ?
8 rZblkans7 Zpopopop6 pZ0Z0Z0Z5 Z0Z0Z0Z04 0Z0Z0Z0Z3 Z0ZnO0Z02 PO0ONOPO1 SNAQJBZR
a b c d e f g h
![Page 154: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/154.jpg)
1 c4
8 rmblkans7 opopopop6 0Z0Z0Z0Z5 Z0Z0Z0Z04 0ZPZ0Z0Z3 Z0Z0Z0Z02 PO0OPOPO1 SNAQJBMR
a b c d e f g h
![Page 155: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/155.jpg)
1 c4 Na6
8 rZblkans7 opopopop6 nZ0Z0Z0Z5 Z0Z0Z0Z04 0ZPZ0Z0Z3 Z0Z0Z0Z02 PO0OPOPO1 SNAQJBMR
a b c d e f g h
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1 c4 Na6 2 c5
8 rZblkans7 opopopop6 nZ0Z0Z0Z5 Z0O0Z0Z04 0Z0Z0Z0Z3 Z0Z0Z0Z02 PO0OPOPO1 SNAQJBMR
a b c d e f g h
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1 c4 Na6 2 c5 NXc5
8 rZblkans7 opopopop6 0Z0Z0Z0Z5 Z0m0Z0Z04 0Z0Z0Z0Z3 Z0Z0Z0Z02 PO0OPOPO1 SNAQJBMR
a b c d e f g h
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1 c4 Na6 2 c5 NXc5 3 e3
8 rZblkans7 opopopop6 0Z0Z0Z0Z5 Z0m0Z0Z04 0Z0Z0Z0Z3 Z0Z0O0Z02 PO0O0OPO1 SNAQJBMR
a b c d e f g h
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1 c4 Na6 2 c5 NXc5 3 e3 a6
8 rZblkans7 Zpopopop6 pZ0Z0Z0Z5 Z0m0Z0Z04 0Z0Z0Z0Z3 Z0Z0O0Z02 PO0O0OPO1 SNAQJBMR
a b c d e f g h
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1 c4 Na6 2 c5 NXc5 3 e3 a6 4 Ne2
8 rZblkans7 Zpopopop6 pZ0Z0Z0Z5 Z0m0Z0Z04 0Z0Z0Z0Z3 Z0Z0O0Z02 PO0ONOPO1 SNAQJBZR
a b c d e f g h
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1 c4 Na6 2 c5 NXc5 3 e3 a6 4 Ne2 Nd3
8 rZblkans7 Zpopopop6 pZ0Z0Z0Z5 Z0Z0Z0Z04 0Z0Z0Z0Z3 Z0ZnO0Z02 PO0ONOPO1 SNAQJBZR
a b c d e f g h
![Page 162: Scacchi e Matematica Mathesis - 150.217.34.175150.217.34.175/files/Dolfi2009-10.pdfStimadelNumerodiPartite;C.Shannon(1950) Numero“tipico” dimosse: circa30(DeGroot,1946: mediasuun](https://reader035.vdocumenti.com/reader035/viewer/2022081523/5fe3708dc78b101bf36a6338/html5/thumbnails/162.jpg)
Bibliografia
• J. Watkins: Across the board: the mathematics of chessboardproblems, Princeton University Press, 2004
• M. Petković: Mathematics and Chess, Dover Publications,1997.
• A. Yaglom e I. Yaglom: Challenging mathematical problemswith elementary solutions: combinatorial analysis andprobability theory, Vol. 1, Holden-Day, 1964.
• N. Elkies e R. Stanley: The mathematical Knight, Math.Intelligencer 25, n.1, (2003), 22-34.
• G. Fricke e altri: Combinatorial problems on chessboards: abrief survey, Graph Theory, Combinatorics and Applications 1(1995), 507–528.
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Pagine Web
• http://en.wikipedia.orgRicca fonte di informazioni (di tipo storico e generale):segnaliamo in particolare le pagine sui campioni (Steinitz,Lasker, ...).
•http://www.permutationpuzzles.org/chess/math_chess.htmlInformazioni e links ad altre risorse
• http://www.math.harvard.edu/∼elkies/chess.htmlHome page di Noam Elkies, ad Harvard.
• http://www.northnet.org/weeks/SoSGiochi su varie superfici (toro, bottiglia di Klein, Nastro diMoebius, ecc...); collegato al libro The Shape of Space.