**Matematinis šachmatų uždavinys** – matematikos uždavinys, kurio užduoties įvykdymui naudojama šachmatų lenta bei figūros. Šiais uždaviniais domisi daugiau matematikai negu šachmatininkai.

Matematika ir šachmatai
-----------------------

Matematika ir šachmatai eilėje sričių yra susiję: matematiko, ar šachmatininko mąstymas panašus, bet matematikų susidomėjimas ne dėl to – jie čia turi savų interesų.
Šachmatų lenta, figūros ir pats žaidimas matematikams tinka, ne tik tikrinant matematikos teiginius ir skaičiavinų formules, bet ir, eksperimentuojant su šachmatų matematiniais uždaviniais sukurtų formulių, panaudojimu kitose srityse.

Šachmatų žaidimo terminai sutinkami kibernetikos, žaidimo teorijos, informatikos, skaičių teorijos, kombinatorikos, grafų teorijos srityse. Pastaroji sritis yra susijusi su, pav. žirgo ėjimo, aštuonių valdovių, figūrų dominavimo ir kt. matematiniais šachmatų uždaviniais.

Šachmatų lenta su išdėstytomis ant jos figūromis ir figūrų ėjimai pasirodė tinkamas modelis, pagimdęs eilę matematinių uždavinių, kuriais domėjosi bei nagrinėjo matematikai:, Leonardas Oileris (*Leonhard Euler* (1707–1783), Adrianas Ležandras (*Adrien-Marie Legendre* (1752–1833))
ir Karlas Frydrichas Gausas (*Johann Carl Friedrich Gauss* (1777–1855)) ir kt.

Istorija
--------

Šachmatų matematiniai galvosūkiai žinomi nuo XIX a. 875 m. jau paminėta žirgo kelionė Rudratos (Rudrata) sanskrito kalba rašytame kūrinyje KAVYALANKARA.

### Šachmatų lentos apėjimas žirgu

Žirgo kelionė aplankant kiekvieną laukelį Žirgo kelionė aplankant kiekvieną laukelį
Leonardo Oilerio surastas šachmatų lentos simetriškas apėjimas žirgu 
Visų šachmatų lentos langelių apėjimo žirgu, neužeinant du kartus į tą patį langelį, tema minima ir 1694 m. išleistojoje prancūzų matematiko Žako Ozanamo (*Jacques Ozanam* (1640–1718) „*Récréations Mathématiques et Physiques*" knygoje. Tai buvo rinkinys pagal 1612 m. pasirodžiusio kito prancūzų matematiko Klūdo Bešės (*Claude Gaspard Bachet de Méziriac* (1581–1638) *Problèmes Plaisans ir Délectable* pavyzdį. Jame buvo pateikta daugybė, skirtų pramogoms ir švietimui socialiniuose susibūrimuose: galvosūkių, gudrybių, įdomiosios matematikos bei mokslinio populiarinimo publikacijų.

Šiuolaikinis šachmatų žirgo uždavinio nagrinėjimas, pradėtas XVIII a. nebežinant apie viduramžių darbus, išskyrus P. Guarini (*Paulo Guarini di Forli*) pusės šachmatų lentos apėjimą žirgu.

Šveicarų matematikas Leonardas Oileris (*Leonhard Euler* (1707–1783), šachmatų žirgo tema dirbo kelerius metus pakol 1757 m. laiške vokiečių matematikui Kristianui Goldbachui (*Christian Goldbach* (1690–1764) pateikė simetrišką lentos apėjimą žirgu, o 1759 m. Berlyne, mokslų akademijos leidinyje *Mémoires de l’Academie Royale des Sciences et Belles Lettres*, paskelbė *įdomaus klausimo, kuris atrodo nebuvo nagrinėtas, sprendimas* (*"Solution d’une question curieuse qui ne paroit soumise à aucune analyse"*) rašytą prancūzų kalba, analizuojantį šachmatų žirgo kelionę po šachmatų lenta.

Tai turėjo įtakos tolesniems temos nagrinėjimui, kuriuos tęsė: italų leidėjas Lelio dalla Volpė (*Lelio dalla Volpe* (1685–1749) 1766 m. pirmą kartą, vietoj skaitmeninės formos, pateikęs eilę *žirgo kelionės* diagramų.

Prancūzų muzikas, chemikas ir matematikas Aleksandras Vandermondas (*Alexandre-Theophile Vandermonde* (1735–1796)) – 1771 m. italų istorikas Kozimo Alesandras Kolinis (*Cosimo Alessandro Collini* (1727–1806)) savo knygoje *Solution du Problème du Cavalier au Jeu des Echecs*, 1773 m. išleistoje Manheime, skelbė, radęs žirgo kelionės uždavinio sprendimą, H. C. fon Varnsdorfas (*H.C. von Warnsdorff*) savo 1823 m. išleistoje knygoje *Des Rösselsprung einfachste und allgemeinste Lösung* pasiūlė taisyklę, kaip išspręsti žirgo uždavinį. , tačiau yra ir abejojančių.

### Aštuonių valdovių uždavinys

Aštuonių valdovių uždavinį pirmas pasiūlė Maksas Bezelis (*Max Friedrich Wilhelm Bezzel* (1824–1871), patalpinęs 1848 m. jį Berlyno laikraštyje *Schachzeitung*. Pirmą kartą visus sprendinius (92) rado Francas Naukas (*Franz Nauck*) 1850 m. paskelbęs apie tai Leipcigo laikraštyje *Illustrierte Zeitung*, o 1874 m. Džeimsas Gleišeris (*James Whitbread Lee Glaisher* (1848–1928)) įrodė, kad kitų sprendimų nėra.

Užduotys
--------

Uždavinių užduotys susijusios:

su šachmatų lenta,

su įvairių figūrų nepersikertančiais maršrutais,

jų išdėstymo lentoje savybėmis.

Yra uždavinių, kuriuose reikia:

sustatyti lentoje didžiausią figūrų skaičių taip, kad jos negrėstų viena kitai,

arba mažiausią figūrų skaičių taip, kad jos atakuotų visus kitus lentoje laukelius.

Yra uždavinių, kuriuose sprendimo esmę sudaro figūrų perstatymas, arba jų keitimas vietomis.

XIX a. buvo populiarūs uždaviniai: apie 8 valdoves, visos šachmatų lentos apėjimą žirgu, apie neliečiamą karalių ir eilė kitų.

### Nepriklausomos figūros

#### Apie 8 valdoves

Aštuonios valdovės. Dvylika pozicijųAštuonios valdovės. Dvylika pozicijų
Prašoma išdėstyti ant šachmatų lentos 8 valdoves taip, kad jos negrasintu viena kitai (t. y. nei viena valdovė neturėtų stovėti vienoj vertikalėje, horizontalėje ar diagonalėje su bet kuria kita valdove) ir nustatyti keliais būdais tai galima būtų padaryti.

Bet kokiu atveju viena valdovė būtinai turi stovėti a4 laukelyje, arba simetriškuose jam laukeliuose 45, d8, e8, h5, h4, e1, d1. Unikalių pozicijų, kurių negalima sudaryti pasukimais, ar pagal veidrodinius atspindžius, yra tik 12-ka. Likusios padėtys – dvyniai, kurie iš 12-kos unikaliųjų, susidaro, pasukus jas pagal ašį 90, 180 ir 270 laipsnių kampu ir pridėjus veidrodinius atspindžius. Pirma pozicija turi 3 dvynius, likusios po 7. Šis uždavinys naudojamas programavimo mokymo vadovėliuose

#### Apie kitas figūras

Kai uždavinyje keliama tokia pat užduotis ir kitoms šachmatų figūroms, tai didžiausias karalių skaičius yra 16, valdovių – 8, bokštų 8, rikių – 14 žirgų 32. Būdų kaip išdėstomos figūros šachmatų lentoje skiriasi: bokštai gali būti išdėliojami vienoje, ar keliose, įstrižainėse 40320 būdais, žirgai tik dviem (ant baltų arba juodų laukelių) ir t. t.

Unikali yra enų matematiko Henrio Dudenio (*Henry Ernest Dudeney* (1857–1930)) šachmatų pozicija, kurioje patalpintos 8-ios valdovės, 8-ni – bokštai, 14-ka – rikių, 21-as žirgas. Joje figūros patalpintos taip, kad savosioms figūroms negrasina.

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Šachmatų figūrų nepriklausomumo uždaviniai| Rodikliai | Valdovė | Bokštas | Žirgas | Rikis | Karalius |
| --- | --- | --- | --- | --- | --- |
| Maksimalus nepuolamų figūrų skaičius 8×8 lentoje | 8 | 8 | 32 | 14 | 16 |
| Figūrų išdėliojimo būdai | 92 | 40320 | 2 | 256 | 281571 |
|

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  **Nepriklausomos figūros** 

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| --- |
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|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

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| 32 taikūs žirgai. |

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  **Nepriklausomos figūros** 

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| --- |
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| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

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|  |

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| 14 nepriklausomų rikių. |

 |  | 
  **Nepriklausomos figros** 

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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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|  |
| --- |
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|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

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| Nepriklausomi bokštai |

 |  | 
  **Nepriklausomos figūros** 

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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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| --- |
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| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

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| 16-ka taikių karalių. |

 |  |  | 
  **Nepriklausomos figūros** 

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| Djudenio uždavinys - 8 valdovės, 8 bokštai, 14 rikių ir 21 žirgas |

 |

 |

### Žirgo kelias

Siūloma, pirmuoju ėjimu, pastačius žirgą ant bet kurio šachmatų lentos laukelio, nuosekliai pereiti su juo visus laukelius, neužimant nei vieno iš jų du kartus. Jei po 65 ėjimo žirgas gali patekti į pradinį laukelį toks žirgo kelias vadinamas uždaru.
Jei surasti žirgo maršrutą sąlyginai yra nesudėtinga, tai apskaičiuoti galimų maršrutų skaičiaus nepavyksta. Žinoma tik tiek, kad sprendinių skaičius viršija 30 milijonų.

### Neliečiamas karalius

Baltųjų karalius stovi šachmatų lentos c3 (arba c6, f6, f3) laukelyje ir valdovė bet kokiam laukelyje. Juodieji karalių. Ar visada baltieji gali nejudindami savo karaliaus užmatuoti juodųjų karalių. Sprendimą, panaudoję komiuterį surado A.L Brudno ir I.J. Landau. Matas esant bet kokiai juodųjų karaliaus ir baltųjų valdovės padėčiai, skelbiamas ne vėliau, kaip per 23 ėjimus. Kai baltųjų karaliaus padėtis kita, o juodųjų karalius bet kur, nejudinant baltųjų karaliaus, mato juodų karaliui paskelbti negalima.

### Figūrų dominavimas

Dominavimo šachmatų uždaviniai siūlo rasti mažiausią vienvardžių figūrų (valdovių, bokštų, rikių, žirgų ir karalių) skaičių, kurį galima išdėlioti ant šachmatų lentos taip, kad jos atakuotų visus neužimtus lentos laukelius.

Dominavimui reikalingas figūrų skaičius nevienodas ir priklauso nuo figūros tipo. Mažiausias dominuojančių valdovių skaičius ant 8x8 (įprastinės) , 9x9, 10x10 ir 11x1 laukelių lentų – penkios, o kitų figūrų ant 8x8 šachmatų lentos daugiau: bokštų – 8, karalių – 9, žirgų 12, rikių – 8 (4 baltaspalvių + 4 juodaspalvių).

Apskaičiuota, kad penkias valdoves dominavimui ant 8x8 laukelių lentos galima išdėlioti 4860, o aštuonis rikius – 20736 būdais.

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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 
 **Figūrų dominavimas** 

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| Penkios dominuojančios valdovės. |

 |  | 
 **Figūrų dominavimas** 

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| Mažiausias dominuojančių žirgų skaičius – 12-ka. 
  |

 |  | 
 **Figūrų dominavimas** 

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| Mažiausias dominuojančių bokštų skaičius - 8. |

 |  | 
  **Figūrų dominavimas** 

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| Mažiausias dominuojančių karalių skaičius- 9. |

 |  | 
  **Figūrų dominavimas** 

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| 7-ios figūros dominuojančios šachmatų lentoje. |

 |

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Dominavimo uždaviniuose galima būti reikalaujama, kad figūros saugotų ne tik laisvuosius laukelius, bet ir užimtuosius, t. y. visus. Tokios figūros vadinamos *sargybinėmis*. Aštuoni bokštai, stovintys vertikalėje, ar horizontalėje sergėja visus šachmatų lentos laukelius. Kitų figūrų visos lentos apsaugai reikia truputi daugiau negu dominavimui: žirgų ir rikių dviem, atitinkamai 14-kos ir 10-ies, o karalių trim, t. y. 12-kos.

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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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  **Figūros - sargybiniai** 

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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

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| Mažiausias **valdovių** skaičius, saugantis visus laukelius - 5. |

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  **Figūros - sargybiniai** 

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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

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| Mažiausias **bokštų** skaičius saugantis visus laukelius -8. |

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  **Figūros - sargybiniai**

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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

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| Mažiausias **žirgų** skaičius, saugantis visus laukelius - 14. |

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  **Figūros - sargybiniai**

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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

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| Mažiausias **rikių** skaičius, saugantis visus laukelius -10. |

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  **Figūros - sargybiniai** 

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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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| --- |
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| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

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| Mažiausias **karalių** skaičius, saugantis visus laukelius - 12. |

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Penkios valdovės būtinai reikalingos jų dominavimui ant šachmatų lentos, bet dvi valdoves galima pakeisti silpnesnėmis figūromis – dviem bokštais, ar net bokštu ir rikiu. Pirmuoju atveju atakuojami visi lentos laukeliai (taip pat ir užimtieji.)

Aštuonios šachmatų figūros gali dominuoti visoje šachmatų lentoje tik tuo atveju, kai rikiai yra vienaspalviai. Jei jie skirtingų spalvų bent vienas lentos laukelis lieka nesaugomas.

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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 
  **Kelios figūros**

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| **A** **Trys** valdovės ir **du** bokštai. |

 |  | 
  **Kelios figūros**

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| **B.** **Trys** valdovės, bokštas ir rikis. |

 |  | 
  **Kelios figūros**

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| **C. Keturios** valdovės ir žirgas |

 |  | 
  **Kelios figūros**

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| **D. Keturios** valdovės ir karalius. |

 |  | 
  **Kelios figūros**

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |
| --- |
|  |
|  | 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| a8 | b8 | c8 | d8 | e8 | f8 | g8 | h8 |
| a7 | b7 | c7 | d7 | e7 | f7 | g7 | h7 |
| a6 | b6 | c6 | d6 | e6 | f6 | g6 | h6 |
| a5 | b5 | c5 | d5 | e5 | f5 | g5 | h5 |
| a4 | b4 | c4 | d4 | e4 | f4 | g4 | h4 |
| a3 | b3 | c3 | d3 | e3 | f3 | g3 | h3 |
| a2 | b2 | c2 | d2 | e2 | f2 | g2 | h2 |
| a1 | b1 | c1 | d1 | e1 | f1 | g1 | h1 |

 |  |
|  |

 |
| **E.** Valdovė, du bokštai, du rikiai, du žirgai. |

 |

 |

Dominavimo uždaviniuose naudojamos ir pasakiškosios figūros: svirplys, bokštinis svirplys, nakviša ir kt.

Šaltiniai
---------

1. ↑  „The knight's tour" (anglų). Nuoroda tikrinta 2019 m. spalio 1 d..
2. ↑  „Origin of chess" (anglų). Nuoroda tikrinta 2019 m. rugsėjo 29 d..
3. ↑  „Knight's Tours on the Half Chessboard (1)" (anglų). Suarchyvuotas originalas 2019-10-01. Nuoroda tikrinta 2019 m. spalio 1 d.. `{{cite web}}`: Nežinomas parametras `|dead-url=` ignoruotas (`|url-status=` suggested) (pagalba)
4. ↑  „Rediscovery of the Knight's Problem 1725 – 1825" (anglų). Suarchyvuota iš originalo 2009-05-05. Nuoroda tikrinta 2019 m. rugsėjo 27 d..`{{cite web}}`: CS1 priežiūra: netinkamas URL (link)
5. ↑  „Rediscovery of the Knight's Problem" (anglų). Nuoroda tikrinta 2019 m. rugsėjo 30 d.. `{{cite web}}`: Cite has empty unknown parameter: `|1=` (pagalba)
6. ↑  „Visi aštuonių valdovių uždavinio sprendimai" (PDF) (anglų). Nuoroda tikrinta 2019 m. rugsėjo 29 d..
7. ↑  Antanas Vilkauskas. Šachmatų kompozicijos pagrindai. Vilnius: Mintis, 2002.— P. 119.
8. ↑  „Математика на шахматной доске" (rusų). Nuoroda tikrinta 2019 m. spalio 8 d..
9. ↑  Шахматы. Энциклопедический словарь / гл. ред. А. Е. Карпов. – М.: Советская энциклопедия, 1990. – С.237-238
10. ↑  „Noam D. Elkies, Richard P. Stanley. Šachmatų žirgas ir matematika" (PDF) (anglų). Nuoroda tikrinta 2020 m. spalio 7 d..
11. ↑  Е.Я. Гик. Шахматы и математика. Москва: Наука, 1983. С. 72-74.
12. ↑  „George Jelliss. Figūrų dominavimas šachmatų lentoje" (anglų). Nuoroda tikrinta 2020 m. spalio 7 d..

Išorinės nuorodos
-----------------

* Šachmatų įdomybės[neveikianti nuoroda]
* Žirgo kelionės po šachmatų lenta uždavinio istorija
* Žirgo uždavinys
* Introducing Knight’s Tours
* Early History of Knight’s Tours
* Rediscovery of the Knight’s Problem
* Knight’s Tour. Non-Intersecting Paths