Disposizione di regine su una scacchiera

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Br1
Livello 6
Livello 6
Messaggi: 465
Iscritto il: mer feb 21, 2007 5:53 pm
Località: Bologna

Messaggio da Br1 »

Jumpy94 ha scritto:Credo che il mio quesito sia stato respinto forse perché troppo semplice :( :( ......comunque, per far finta che qualcuno abbia risposto, mi do da solo una soluzione.
Ma non credete di scamparla, aprirò un topic dove metterò uno schema abbastanza difficile........ :twisted: :twisted: :twisted: :twisted:
Ciao, Jumpy!

Sei forte :D

Comunque, sì, penso anch'io che sia meglio aprire un
topic a parte, in modo che ciascuno di noi possa meglio
individuare il problema.

Di solito, in effetti, inserire le proprie proposte, sia pur
interessanti, all'interno di un topic già dedicato a una
questione ben precisa e inoltre ancora aperta, potrebbe
non essere il modo migliore per renderle visibili. E poi
potrebbe essere più difficile rintracciarle in seguito,
soprattutto per quelli - come me - che hanno i loro tempi...

D'altra parte, non so nemmeno dirti se il tuo quiz sia
facile o no, e sai perché? Perché non ho praticamente
mai giocato a scacchi in vita mia :oops:
Ricordo che ci ho provato con mio fratello, di dieci anni
più giovane, ma era così bravo a vincere che in breve
ho cominciato ad annoiarmi.

Non mi sono capitate altre occasioni per riprendere i
tentativi o forse non le ho veramente cercate.

Mah... vedremo.

Buona serata!
Bruno

giobimbo
Livello 5
Livello 5
Messaggi: 343
Iscritto il: sab nov 19, 2005 5:14 pm
Località: Biella

Messaggio da giobimbo »

Come accennato precedentemente, se n = dispari divido la scacchiera in tre parti, superiore, centrale e inferiore; se n = pari la divido in due parti, superiore e inferiore, eccetto che per n = 8 + 6k che ha una costruzione particolare. Per il momento mi limito a poche cose per lasciare piuttosto parlare gli esempi costruttivi, spiegando di più solo su richiesta per le parti non ben comprese (sempre che questa sia la soluzione giusta....).

Nelle miniscacchiere rettangolari le regine sono disposte a mossa di cavallo, con due distinte configurazioni, in fila o in doppia fila, come da esempio sotto, ma comunque stanno sempre nelle colonne dispari o sempre nelle colonne pari:

0...0...0...0...0...0...0...0...0...0...0...1
0...0...0...0...0...0...0...0...0...1...0...0
0...0...0...0...0...0...0...1...0...0...0...0
0...0...0...0...0...1...0...0...0...0...0...0
0...0...0...1...0...0...0...0...0...0...0...0
0...1...0...0...0...0...0...0...0...0...0...0 Regine in fila

0...0...0...0...0...0...0...0...1...0...0
0...0...0...0...0...0...0...0...0...0...1
0...0...0...0...1...0...0...0...0...0...0
0...0...0...0...0...0...1...0...0...0...0
1...0...0...0...0...0...0...0...0...0...0
0...0...1...0...0...0...0...0...0...0...0 Regine in doppia fila

Infine, se la larghezza n = dispari le regine in fila sono disposte in modo centrato, ovvero le regine che stanno alle estremità sinistra e destra, sono poste alla stessa distanza sia dal lato sinistro che dal lato destro della scacchiera rettangolare.


Passiamo agli esempi pratici. Sia n = 4, 10, 16, ..., allora:

0...0...1...0
1...0...0...0
0...0...0...1
0...1...0...0

0...0...0...0...0...0...0...0...1...0
0...0...0...0...0...0...1...0...0...0
0...0...0...0...1...0...0...0...0...0
0...0...1...0...0...0...0...0...0...0
1...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...1
0...0...0...0...0...0...0...1...0...0
0...0...0...0...0...1...0...0...0...0
0...0...0...1...0...0...0...0...0...0
0...1...0...0...0...0...0...0...0...0

0...0...0...0...0...0...0...0...0...0...0...0...0...0...1...0
0...0...0...0...0...0...0...0...0...0...0...0...1...0...0...0
0...0...0...0...0...0...0...0...0...0...1...0...0...0...0...0
0...0...0...0...0...0...0...0...1...0...0...0...0...0...0...0
0...0...0...0...0...0...1...0...0...0...0...0...0...0...0...0
0...0...0...0...1...0...0...0...0...0...0...0...0...0...0...0
0...0...1...0...0...0...0...0...0...0...0...0...0...0...0...0
1...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...1
0...0...0...0...0...0...0...0...0...0...0...0...0...1...0...0
0...0...0...0...0...0...0...0...0...0...0...1...0...0...0...0
0...0...0...0...0...0...0...0...0...1...0...0...0...0...0...0
0...0...0...0...0...0...0...1...0...0...0...0...0...0...0...0
0...0...0...0...0...1...0...0...0...0...0...0...0...0...0...0
0...0...0...1...0...0...0...0...0...0...0...0...0...0...0...0
0...1...0...0...0...0...0...0...0...0...0...0...0...0...0...0

Regine in fila sia sotto che sopra, dunque; lo stesso per n = 6, 12, ..., per cui disegno solo la scacchiera di partenza:

0...0...0...0...1...0
0...0...1...0...0...0
1...0...0...0...0...0
0...0...0...0...0...1
0...0...0...1...0...0
0...1...0...0...0...0

Per n = 8, 14, 20, ..., abbiamo regine in doppia fila nella parte superiore e, indicata col numero 2, una regina ballerina (ma in realtà non lo è) nella prima fila di caselle:

0...0...0...0...0...0...1...0
1...0...0...0...0...0...0...0
0...0...1...0...0...0...0...0
0...0...0...0...0...0...0...1
0...0...0...0...0...1...0...0
0...0...0...1...0...0...0...0
0...1...0...0...0...0...0...0
0...0...0...0...2...0...0...0

0...0...0...0...0...0...0...0...1...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...1...0...0...0
0...0...0...0...1...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...1...0...0...0...0...0...0...0
1...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...1...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...1
0...0...0...0...0...0...0...0...0...0...0...1...0...0
0...0...0...0...0...0...0...0...0...1...0...0...0...0
0...0...0...0...0...0...0...1...0...0...0...0...0...0
0...0...0...0...0...1...0...0...0...0...0...0...0...0
0...0...0...1...0...0...0...0...0...0...0...0...0...0
0...1...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...2...0

0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...1...0
0...0...0...0...0...0...0...0...0...0...0...0...1...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...0...1...0...0...0...0...0
0...0...0...0...0...0...0...0...1...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...1...0...0...0...0...0...0...0...0...0
0...0...0...0...1...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...1...0...0...0...0...0...0...0...0...0...0...0...0...0
1...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...1...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...1
0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...1...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...1...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...1...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...1...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...1...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...1...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...1...0...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...1...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0
0...1...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...2...0...0...0


Passiamo ai lati dispari, n = 5, 11, 17, ...

0...1...0...0...0
0...0...0...0...1
0...0...1...0...0
1...0...0...0...0
0...0...0...1...0

0...0...0...0...1...0...0...0...0...0...0
0...0...1...0...0...0...0...0...0...0...0
1...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...1...0
0...0...0...0...0...0...0...1...0...0...0
0...0...0...0...0...1...0...0...0...0...0
0...0...0...1...0...0...0...0...0...0...0
0...1...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...1
0...0...0...0...0...0...0...0...1...0...0
0...0...0...0...0...0...1...0...0...0...0

0...0...0...0...0...0...0...1...0...0...0...0...0...0...0...0...0
0...0...0...0...0...1...0...0...0...0...0...0...0...0...0...0...0
0...0...0...1...0...0...0...0...0...0...0...0...0...0...0...0...0
0...1...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...1
0...0...0...0...0...0...0...0...0...0...0...0...0...0...1...0...0
0...0...0...0...0...0...0...0...0...0...0...0...1...0...0...0...0
0...0...0...0...0...0...0...0...0...0...1...0...0...0...0...0...0
0...0...0...0...0...0...0...0...1...0...0...0...0...0...0...0...0
0...0...0...0...0...0...1...0...0...0...0...0...0...0...0...0...0
0...0...0...0...1...0...0...0...0...0...0...0...0...0...0...0...0
0...0...1...0...0...0...0...0...0...0...0...0...0...0...0...0...0
1...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...0...0...1...0
0...0...0...0...0...0...0...0...0...0...0...0...0...1...0...0...0
0...0...0...0...0...0...0...0...0...0...0...1...0...0...0...0...0
0...0...0...0...0...0...0...0...0...1...0...0...0...0...0...0...0

Regine in fila sopra, in mezzo e sotto, quindi; lo stesso per n = 7, 13, ..., per cui disegno solo la scacchiera di partenza:

0...0...1...0...0...0...0
1...0...0...0...0...0...0
0...0...0...0...0...1...0
0...0...0...1...0...0...0
0...1...0...0...0...0...0
0...0...0...0...0...0...1
0...0...0...0...1...0...0

Per n = 9, 15, 21, ..., abbiamo regine in doppia fila sopra e sotto, ma la disposizione al crescere delle scacchiere è semplice, per cui mi limito ai primi due casi:

0...1...0...0...0...0...0...0...0
0...0...0...1...0...0...0...0...0
0...0...0...0...0...0...0...0...1
0...0...0...0...0...0...1...0...0
0...0...0...0...1...0...0...0...0
0...0...1...0...0...0...0...0...0
1...0...0...0...0...0...0...0...0
0...0...0...0...0...1...0...0...0
0...0...0...0...0...0...0...1...0

0...0...0...0...1...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...1...0...0...0...0...0...0...0...0
1...0...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...1...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...1...0
0...0...0...0...0...0...0...0...0...0...0...1...0...0...0
0...0...0...0...0...0...0...0...0...1...0...0...0...0...0
0...0...0...0...0...0...0...1...0...0...0...0...0...0...0
0...0...0...0...0...1...0...0...0...0...0...0...0...0...0
0...0...0...1...0...0...0...0...0...0...0...0...0...0...0
0...1...0...0...0...0...0...0...0...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...0...0...1...0...0
0...0...0...0...0...0...0...0...0...0...0...0...0...0...1
0...0...0...0...0...0...0...0...1...0...0...0...0...0...0
0...0...0...0...0...0...0...0...0...0...1...0...0...0...0

Sancho Panza
Livello 4
Livello 4
Messaggi: 151
Iscritto il: gio ott 12, 2006 9:01 pm

Messaggio da Sancho Panza »

Ciao Giobimbo,
sapevo che avresti presto trovato la soluzione.
La tua soluzione è analoga alla mia, (ma con il tuo sistema delle regine in doppia fila hai trovato un sistema più elegante per risolvere il problema)
Desidero fare i complimenti anche a Jepa che comunque aveva trovato
una soluzione valida per i casi (N mod 6)=1 e (N mod 6)=5

A questo punto comincia la parte più interessante del problema:
in quanto anche tu Giobimbo hai trovato una soluzione con struttura che varia modulo 6.
Si tratta ora di dipanare il mistero: questa dipendenza dal modulo 6 da cosa dipende?
E' dovuta al fatto che è più facile trovare una forma generale per le soluzioni in questo modo?
E' possibile trovare una forma generale delle soluzioni che non varii modulo 6?
Perché le maggiori difficoltà si incontrano per N (modulo 6)=2 ?
Eccetera, eccetera...

Se vuoi provare a rispondere queste domande mi fai un piacere, io conto molto sulla tua esperienza nello studio delle permutazioni.
Già che ci sei, volendo potresti provare anche a studiare le analogie tra questo problema e quello delle "Pedine in ordine sparso".
:wink:

ZioGiò
Livello 4
Livello 4
Messaggi: 156
Iscritto il: sab gen 14, 2006 4:30 pm
Località: Mailand (Milano)
Contatta:

Messaggio da ZioGiò »

Hi!
le mie limitate capacità di programmatore non mi permettono di andare oltre
Beh, direi che in breve anche la precisione macchina non ti permetterebbe di andare oltre :)
365596 è un bel numerino... Vuoi dire che la funzione non è monotona crescente? Per n=6 davvero solo 4 soluzioni possibili, (che poi sono speculari e si riducono solo a una soluzione)? E' davvero curioso, anche perchè questo sembra essere un numero importante nella risoluzione del problema... Chissa per N = 36 cosa viene fuori...

Saludos!
"Voi mi considerate un uomo sanza lettere, ma siete degli stolti perché le mie cose sono date dall'esperienza non dalle parole."
Leonardo Da Vinci

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