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Otto Donne PDF Stampa E-mail
domenica 06 maggio 2007
Quante Donne si possono porre sulla scacchiera in modo che non si attacchino vicendevolmente? Il problema fu posto per la prima volta nel 1848 da una rivista tedesca, la "Scachzeitung", a seguito di un quesito da parte del filologo professor Nauck, che lo propose anche a Karl Friedrich Gauss (1777-1855).
Il celebre matematico inizio' allora sull'argomento una fitta corrispondenza con l'astronomo Heinrich Schumacher (1780-1850). Finalmente dopo due anni, servendosi della teoria dei gruppi che proprio in quegli anni muoveva i primi passi, i due trovarono le dodici posizioni base che risolvono il problema e che per rotazione e riflessione della scacchiera portano il totale di possibili soluzioni differenti a 92.
Ma perche' venisse dimostrato che queste 92 soluzioni sono le uniche possibili si dovette attendere fino al 1874, anno in cui l'inglese Gleisher, docente all'universita' di Cambridge (in un articolo apparso su "Philosophical Magazine), e il tedesco Gunther dell'universita' di Lipsia ne diedero una dimostrazione basata sulla teoria dei determinanti.

a4 b1 c5 d8 e2 f7 g3 h6
a4 b1 c5 d8 e6 f3 g7 h2
a4 b2 c5 d8 e6 f1 g3 h7
a4 b2 c7 d3 e6 f8 g1 h5
a4 b2 c7 d3 e6 f8 g5 h1
a4 b2 c7 d5 e1 f8 g6 h3
a4 b2 c8 d5 e7 f1 g3 h6
a4 b2 c8 d6 e1 f3 g5 h7
a4 b6 c1 d5 e2 f8 g3 h7
a4 b7 c5 d2 e6 f1 g3 h8
a4 b8 c1 d5 e7 f2 g6 h3
a4 b6 c8 d2 e7 f1 g3 h5

Ogni soluzione base ne fornisce altre tre per rotazione della scacchiera e quattro per riflessione su uno dei quattro lati. Fa eccezione solo la soluzione 12 che essendo simmetrica da' origine a sole tre altre soluzioni derivate.

Ancora sulla scacchiera di 64 case, ci sono piu' di 4.000 possibilita' per sistemare cinque Donne in modo da controllare tutte le caselle almeno una volta.
Per esempio: a2 c4 d5 e6 g8.

Si possono anche piazzare le otto Donne in modo da far loro controllare il minor numero possibile di caselle della scacchiera, ovvero 53, con sole 11 case non attaccate.
Una possibile soluzione: b1 b2 f2 g1 g3 g7 h2 h7.

Torri, Cavalli, Alfieri
Identica cosa per le Torri. Se ne possono sistemare un massimo di otto senza che si attacchino reciprocamente; una soluzione e' piazzarle lungo le caselle della grande diagonale. Complessivamente le posizioni possibili in cui disporre otto Torri sulla scacchiera in modo che non si attacchino reciprocamente sono 40.320.
Con gli Alfieri il numero sale a 14. Per esempio otto sulla prima traversa e sei sull'ottava (su quest'ultima lasciando libere le due caselle alle estremita').
I Cavalli salgono a 32: basta piazzarli tutti sulle caselle del medesimo colore.

Studi approfonditi su questi 'giochi matematici' furono fatti dal matematico inglese Ernest Dudeney (1857-1930), che tra l'altro si definiva l'ideatore delle parole incrociate.
 
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71 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170918 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170918 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170918 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
72 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170918) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
73 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170919 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170919 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170919 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
74 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170919) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
75 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170920 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170920 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170920 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
76 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170920) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
77 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170921 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170921 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170921 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
78 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170921) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
79 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170922 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170922 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170922 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
80 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170922) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
81 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170923 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170923 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170923 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
82 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170923) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
83 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170924 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170924 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170924 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
84 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170924) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
85 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170925 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170925 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170925 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
86 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170925) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
87 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170926 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170926 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170926 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
88 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170926) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
89 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170927 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170927 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170927 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
90 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170927) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
91 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170928 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170928 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170928 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
92 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170928) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
93 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170929 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170929 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170929 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
94 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170929) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
95 SELECT e.extid, start_date, end_date from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE ( ( DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170930 AND DATE_FORMAT(e.end_date,'%Y%m%d') >= 20170930 ) OR ( DATE_FORMAT(e.start_date,'%Y%m%d') = 20170930 ) ) AND c.published = '1' AND approved = '1' AND recur_type = ''ORDER BY start_date,title ASC
96 SELECT e.extid, recur_type, recur_val, recur_until, start_date, end_date, recur_end_type, recur_count from chess_extcal_events AS e LEFT JOIN chess_extcal_categories AS c ON e.cat=c.cat_id WHERE (DATE_FORMAT(e.start_date,'%Y%m%d') <= 20170930) AND c.published = '1' AND approved = '1' AND recur_type <> '' ORDER BY start_date,title ASC
97 SELECT m.* FROM chess_menu AS m WHERE menutype = 'mainmenu' AND published = 1 AND access <= 0 ORDER BY parent, ordering
98 SELECT m.* FROM chess_menu AS m WHERE menutype = 'usermenu' AND published = 1 AND access <= 0 ORDER BY parent, ordering
99 SELECT COUNT( id ) AS count_users FROM chess_users
100 SELECT COUNT( id ) AS count_items FROM chess_content
101 SELECT COUNT( id ) AS count_links FROM chess_weblinks WHERE published = 1
102 SELECT SUM( hits ) AS count FROM chess_stats_agents WHERE type = 1