❈✉❜❡s ❛♥❞ r❡❧❛t❡❞ ♣r♦❜❧❡♠s ◆✳❱✳▲ñ✐ ❍❛♥♦✐ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ◆❍Ú◆● ●➐ ❑❍➷◆● ❉❸❨ ✣×Đ❈ ▲Ĩ❈ ▲❰◆ ❚❍➐ P❍❷■ ❉❸❨ ◆●❆❨ ❚Ø ◆❍➘✦ ❍ë✐ ❚♦→♥ ❤å❝ ❍➔ ◆ë✐ ❝ị♥❣ ❙ð ●✐→♦ ❞ư❝ ❍➔ ◆ë✐ ❦➳t ❤ñ♣ ①✉➜t ❜↔♥ ❝✉è♥ ❦➾ ②➳✉ ♥❤➙♥ ❞à♣ tê ❝❤ù❝ ❝✉ë❝ t❤✐ ❍❖▼❈ ❧➛♥ t❤ù 15 ✈➔ ❝ô♥❣ ❧➔ ❧➛♥ ✤➛✉ t✐➯♥ ❍❖▼❈ trð t❤➔♥❤ ❝✉ë❝ t❤✐ ố t õ ữợ t ỹ r ú tổ ụ ỹ ữủ ợ t ởt ❜➔✐ ❜→♦ ✈➲ ✏❑❤è✐ ❧➟♣ ♣❤÷ì♥❣ ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✑✳ ✣➙② ❧➔ ♠ët ✤➲ t➔✐ t❤ó ✈à✱ ❤➜♣ ❞➝♥ ✈➔ ❧✉æ♥ ❧✉æ♥ tr➫✳ ❇➔✐ ❜→♦ ♥➔② tê♥❣ ❦➳t ♠ët sè ❝→❝ ❦➳t q✉↔ q✉❛♥ trå♥❣ ❝❤ó♥❣ tỉ✐ ✤❛♥❣ tr♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥✳ ◆❤ú♥❣ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❦❤è✐ ❧➟♣ ♣❤÷ì♥❣ t❤➻ ♥❤✐➲✉✱ ♥❤÷♥❣ ♥❤ú♥❣ t➔✐ ❧✐➺✉ tê♥❣ ❦➳t ✈➔ ♣❤➙♥ ❧♦↕✐ ✈➲ ♥â t❤➻ ❧↕✐ ❝â ❦❤æ♥❣ ♥❤✐➲✉✱ ♠ët ♣❤➛♥ ❝❤➢❝ ❞♦ ❤↕♥ ❝❤➳ ✈➲ ❝æ♥❣ ❝ư t❤➸ ❤✐➺♥ ❝ơ♥❣ ♥❤÷ ❦❤↔ ♥➠♥❣ ①✉➜t ❜↔♥ ♠➛✉✱ ✈➻ ✤➲ t➔✐ ♥➔② ❝➛♥ ♥❤✐➲✉ ❤➻♥❤ ✈➩ ✈➔ ♠➛✉ s➢❝✳ ❚r♦♥❣ ✤➲ →♥ ✤❛♥❣ t❤ü❝ ❤✐➺♥ ♥➔②✱ s♦♥❣ s♦♥❣ ✈ỵ✐ ♠ư❝ ✤➼❝❤ tê♥❣ ❤đ♣✱ ♣❤➙♥ ❧♦↕✐ ✈➔ ❤➺ t❤è♥❣ ❤â❛ ❝→❝ ❦➳t q✉↔ q✉❛♥ trå♥❣ ❝õ❛ ❝→❝ ♥❣❤✐➯♥ ❝ù✉ t➟♣ tr✉♥❣ ✈➲ ❦❤è✐ ❧➟♣ ♣❤÷ì♥❣✱ ❝❤ó♥❣ tỉ✐ ❝á♥ ✤÷❛ ♠ư❝ t✐➯✉✿ ❉➵ ❞↕② ❉➵ ❤å❝ − − ❱➔ ❦❤ì✐ ỗ ự t tr t t r ❣✐↔♥❣ ❞➟② ❝❤ó♥❣ t❛ ❝ơ♥❣ ❝↔♠ t❤➜② ✈➜♥ ✤➲ ♣❤➥♥❣ ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❝â ♠è❝ ♣❤➙♥ ❝→❝❤ ❝❤➼♥❤ ❧➔ ❦❤è✐ ữỡ ữủ ợ s ♥❣❤➽ ❧➟♣ ♣❤÷ì♥❣ t❤➻ ✈✐➺❝ ❝❤✐➳♠ ❧➽♥❤ ♥❤➣♥ q✉❛♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ t♦→♥ ❤✐➺♥ ✤↕✐ trð t❤➔♥❤ ❣➛♥ ❣ơ✐✳ P❤→t ❦✐➳♥ t→♦ ❜↕♦✿ ◆❍Ú◆● ●➐ ❑❍➷◆● ❉❸❨ ✣×Đ❈ ▲Ĩ❈ ▲❰◆ ❚❍➐ P❍❷■ ❉❸❨ ◆●❆❨ ❚Ø ◆❍➘✦ ❈ô♥❣ ❧➔ ✤ë♥❣ ❧ü❝ t❤ỉ✐ t❤ó❝ ❝❤ó♥❣ tỉ✐ ❜➢t t❛② ✈➔♦ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ①➙② ❞ü♥❣ ❣✐→♦ tr➻♥❤ t❤➼ ✤✐➸♠ ♥➔②✳ ❈❤ó♥❣ tỉ✐ ❤② ✈å♥❣ ✤÷đ❝ sü q✉❛♥ t➙♠ ✈➔ ✤â♥❣ ❣â♣ t➼❝❤ ỹ ỗ t sỵ♠ ❝â ♠ët t➔✐ ❧✐➺✉ ❤å❝ ❜ê ➼❝❤✳ ✣➙② ❧➔ ❜➔✐ ❜→♦ ❜➡♥❣ t✐➳♥❣ ❆♥❤✳ ❈❤ó♥❣ tỉ✐ s➩ ❤♦➔♥ ❝❤➾♥❤ t✉②➸♥ t➟♣ ❜➡♥❣ t✐➳♥❣ ❱✐➺t✳ ❳✐♥ ❝↔♠ ì♥ ❝→❝ ❜↕♥✦ ❚❤÷ tø ①✐♥ ❝❤✉②➸♥ ✈➲ ✤à❛ ❝❤➾✿ ▲♦✐s❝❡♥t❡r❅❣♠❛✐❧✳❝♦♠ ✶ ▼ư❝ ❧ö❝ ✶ ❙♣r❡❛❞ t❤❡ ❝✉❜❡ ♦♥ t❤❡ ♣❧❛♥❡ ✸ ✷ ❈✉❜❡s ✺ ✸ ❈✉t t❤❡ ❝✉❜❡ ❜② ❛ ♣❧❛♥❡ ✼ ✹ P❛✐♥t ❈♦❧♦r ❝✉❜❡s ✾ ✺ P❛t❤s ✐♥ t❤❡ ❝✉❜❡ ✶✸ ✻ ❖t❤❡r ❡①❡r❝✐s❡s ✶✹ ❑➌ ❍❖❸❈❍ ◆●❍■➊◆ ❈Ù❯ P❛rt ❆ ✶✳ ❙♣r❡❛❞ t❤❡ ❝✉❜❡ ♥❡t ♦♥ t❤❡ ♣❧❛♥❡ ✷✳ ❈✉❜❡s ✸✳ ❈✉tt✐♥❣ ❛ ❝✉❜❡ ❜② ❛ ♣❧❛♥❡ ✹✳ ❈♦❧♦r✐♥❣ ❝✉❜❡s ✺✳ P❛t❤s ✐♥ ❛ ❝✉❜❡ P❛rt ❇ ✻✳ ●❡♦♠❡tr✐❝ tr❛♥s❢♦r♠❛t✐♦♥s ✇✐t❤ ❝✉❜❡s ✼✳ ❋✐❧❧✐♥❣ ❝✉❜❡s ♦r s♣❛❝❡s ✇✐t❤ ❝✉❜❡✲❧✐❦❡ s❤❛♣❡s ✽✳ Pr♦❜❛❜✐❧✐t②✱ ❞✐❝❡ ❣❛♠❡s ✾✳ ❈❛❧❝✉❧❛t✐♦♥s r❡❧❛t❡❞ t♦ ❝✉❜❡s ✶✵✳ ❖t❤❡r ♣r♦❜❧❡♠s ✷ ❈✉❜❡s ❛♥❞ r❡❧❛t❡❞ ♣r♦❜❧❡♠s ❜② ◆✳❱✳▲ñ✐ ❍❛♥♦✐ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ▼❛t❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠s ❛r♦✉♥❞ ❝✉❜❡s ❛r❡ ✈❡r② ♦❧❞ ❜✉t r❡♠❛✐♥ ✐♥t❡r❡st✐♥❣ t♦ t❤✐s ❞❛②✳ ❚❤❡② ♣♦s❡ ❡①❝✐t✐♥❣ ❝❤❛❧❧❡♥❣❡s t♦ ❧❡❛r♥❡rs ❛♥❞ ❡❞✉❝❛t♦rs ❛❧✐❦❡✳ ❚❤❡s❡ ♣r♦❜❧❡♠s s❡t ❛♣❛rt ❣♦♦❞ st✉❞❡♥ts ❢r♦♠ t❤❡ ❛✈❡r❛❣❡✳ ■♥ t❤✐s ❞♦❝✉♠❡♥t✱ ■ tr② t♦ s✉♠♠❛r✐③❡ ❛♥❞ ❝❛t❡❣♦r✐③❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣r♦❜❧❡♠s r❡✈♦❧✈✐♥❣ ❛r♦✉♥❞ ❝✉❜❡s✳ ❘❡❢❡r❡♥❝❡ ♠❛t❡r✐❛❧ ❝❛♥ ❜❡ ❢♦✉♥❞ ❛t ❬✶❪✱ ❬✷❪✱ ❬✸❪✱ ❬✹❪✱ ❬✺❪✳ ❆ ♥✉♠❜❡r ♦❢ ♣r♦❜❧❡♠s ✇❡r❡ ❝♦❧❧❡❝t❡❞ ❢r♦♠ ❍✉♥❣❛r✐❛♥ ♠❛t❤❡♠❛t✐❝s ❝❛♠♣s✳ ✶ ❙♣r❡❛❞ t❤❡ ❝✉❜❡ ♦♥ t❤❡ ♣❧❛♥❡ ✶✳✶✳ ❲❡ ✇✐❧❧ s❤♦✇ t❤❛t 11 ❢❧❛t ♥❡ts ❝❛♥ ❜❡ ❞r❛✇♥ ❢♦r ❛ ❝✉❜❡✳ ❈❛s❡s t❤❛t ❝❛♥ ❜❡ r♦t❛t❡❞ t♦ ♦✈❡r❧❛♣ ❛r❡ ♥♦t ❝♦✉♥t❡❞ ❛s ❞✐❢❢❡r❡♥t✳ ❲❡ ❝❛♥ ❧✐st t❤❡ s❡♣❛r❛t❡ ♥❡ts✳ ❛✮ ❚❤❡r❡ ❛r❡ ❢♦✉r sq✉❛r❡s ✐♥ ❛ ❧✐♥❡✳ ❇❡❝❛✉s❡ ✐❢ t❤❡r❡ ✇❡r❡ sq✉❛r❡s ✐♥ ♦♥❡ ❧✐♥❡✱ ✐t ❝♦✉❧❞ ♥♦t ❜❡ ❢♦❧❞❡❞ ✐♥t♦ ❛ ❝✉❜❡ ❛s t✇♦ ❢❛❝❡s ✇♦✉❧❞ ♦✈❡r❧❛♣✳ ❚❤✐s ❝❛s❡ ♣r♦✈✐❞❡s s✐① ❞✐❢❢❡r❡♥t ♥❡ts ❢r♦♠ t♦ 6✳ ❜✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② ❋✐❣✳ 7✱ 8✱ ❛♥❞ ❛ sq✉❛r❡s ✐♥ ❛ r♦✇✳ ❲❡ ❣❡t t❤❡ 3−❧✐♥❡ 2−❧✐♥❡ ✭3, 3✮ ♥❡t ✐♥ ❋✐❣✉r❡ 10✳ ✸ ♥❡ts ✭2, 3, 1✮ ♦❢ t❤❡ sq✉❛r❡s ✐♥ ❝✮ ❋✐♥❛❧❧②✱ t❤❡r❡ ✐s ❛ 3−❧✐♥❡ ✭2, 2, 2✮ sq✉❛r❡ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ 11✳ ❤❡ ❛❜♦✈❡ ♥❡t ✇❡ ❝❛♥ ❢♦❧❞ t♦ ❣❡t t❤❡ ❝✉❜❡✳ ✶✳✷ ❈✉t ❛ ❝✉❜❡ ❛❧♦♥❣ t❤❡ ❡❞❣❡s t♦ ❣❡t t❤❡ ❝✉❜❡ ♥❡t✳ ●✉✐❞❡✿ ❖♥ t❤❡ ♣✐❝t✉r❡✳ ❊①❡r❝✐s❡ ✶✳✶✳ ❈✉t t❤❡ ❝✉❜❡ ❛❧♦♥❣ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡❞❣❡s t♦ ❣❡t t❤❡ ❝✉❜❡s ❢r♦♠ ❊①❡r❝✐s❡ ✶✳✷✳ ❖❢ A, B, C, D ✇❤✐❝❤ ❝✉❜❡ ♠❛t❝❤❡s t❤❡ ♥❡t s❤♦✇❡❞ ♦♥ t❤❡ ❧❡❢t❄ ✹ t♦ 11✳ ❊①❡r❝✐s❡ ✶✳✸✳ ❖♥ t❤❡ ♣✐❝t✉r❡✱ ❡❛❝❤ ♥❡t ✐s ♠❛❞❡ ♦❢ sq✉❛r❡s ❛♥❞ tr✐❛♥❣❧❡s✳ ❲❤✐❝❤ ♥❡ts ❝❛♥ ❜❡ ❢♦❧❞❡❞ ✐♥t♦ ❛ ❝✉❜❡❄ ❊①❡r❝✐s❡ ✶✳✹✳ ❨♦✉ ❤❛✈❡ ❛ sq✉❛r❡ ♣❛♣❡r✳ ❍❛✈❡ s❤♦✉❧❞ ②♦✉ ❝✉t ♦♥❡ ♣✐❡❝❡ ❢r♦♠ ✐t s♦ t❤❛t ✐t ❝❛♥ ❜❡ ❢♦❧❞❡❞ ✐♥t♦ ❛ ❝✉❜❡ ♦❢ t❤❡ ❧❛r❣❡st ✈♦❧✉♠❡❄ ❨♦✉ ❝❛♥ ❢✐♥❞ ♠♦r❡ ❡①❡r❝✐s❡s ✐♥ ❘❡❢❡r❡♥❝❡ ❬✸❪✱ ❬✹❪✳ ✷ ❈✉❜❡s Pr♦❜❧❡♠ ✷✳✶✳ ❈❛♥ ②♦✉ ❝✉t ❛ ❝✉❜❡ ✐♥t♦ 20 ❝✉❜❡s❄ ❈❛♥ ②♦✉ ❝✉t ✐t ✐♥t♦ 50 ❝✉❜❡s❄ ❙♦❧✉t✐♦♥✿ P♦ss✐❜❧❡ ❢♦r ❜♦t❤ ❝❛s❡s✳ × × 2✱ × × 3✱ 64 → 1, 27 → 1, → 1✳ ❲✐t❤ t❤❡ ❞✐♠❡♥s✐♦♥s r❡✈❡rs❡✱ ✐t ✐s ❛♥❞ × × 4✱ ✇❡ ❝❛♥ ❝r❡❛t❡ ❚❤❡r❡❢♦r❡✿ 20 = 27 − (8 − 1) = 27 − 50 = 64 − · (8 − 1) = 64 − · ✺ 8✱ 27✱ ❛♥❞ 64 ❝✉❜❡s✳ ■♥ Pr♦❜❧❡♠ ✷✳✷✳ ❈❛♥ ②♦✉ ❝✉t ❛ ❝✉❜❡ ✐♥t♦ ❆♥s✇❡r✿ P♦ss✐❜❧❡ 48 s♠❛❧❧❡r ❝✉❜❡s❄ 27 + · (8 − 1) = 27 + · = 48✳ Pr♦❜❧❡♠ ✷✳✸✳ ❈❛♥ ②♦✉ ❝✉t t❤❡ ❝✉❜❡ ✐♥t♦ 49 ❝✉❜❡s❄ ❆♥s✇❡r✿ P♦ss✐❜❧❡✳ ✉♥✐ts ✭6 × × 6✮✳ ❉✐✈✐❞❡ ✐t ✐♥t♦ × = 36 ✉♥✐t ✭1 × × 1✮ ✭2 × × 2✮ ❛♥❞ × = ❝✉❜❡s ✭3 × × 3✮✳ ❚❤❡♥ 36 + + = 49✳ ❯s❡ ❛ ❝✉❜❡ ✇✐t❤ ❢❛❝❡s ♦❢ ·3 × = ❝✉❜❡s✱ ❊①❡r❝✐s❡ ✷✳✶✳ ❋♦r ✇❤❛t n ✈❛❧✉❡ ❝❛♥ ②♦✉ ❝✉t ❛ ❝✉❜❡ ✐♥t♦ n ❝✉❜❡s✱ s♠❛❧❧ ❝✉❜❡s❄ Pr♦❜❧❡♠ ✷✳✹✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ❝✉t ❛ ❝✉❜❡ ✐♥t♦ ✐❞❡♥t✐❝❛❧ ♣②r❛♠✐❞s❄ ❈❛♥ ✐t ❜❡ ❝✉t ✐♥t♦ ✐❞❡♥t✐❝❛❧ ♣②r❛♠✐❞s❄ ❙♦❧✉t✐♦♥✿ ❛✮ P♦ss✐❜❧❡✳ ❚❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✉❜❡ ❝♦♥♥❡❝t❡❞ t♦ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♠❛❦❡s s✐① ♣②r❛♠✐❞s✳ ❜✮ P♦ss✐❜❧❡✳ ❈♦♥♥❡❝t t❤❡ ✈❡rt✐❝❡s ♦♥ ♦♥❡ ❢❛❝❡ ✇✐t❤ t❤❡ ♦t❤❡r ✈❡rt✐❝❡s ♦♥ ♦t❤❡r ❢❛❝❡s ❛♥❞ ❝♦♥t✐♥✉❡ ✉♥t✐❧ ②♦✉ ❣❡t ✐❞❡♥t✐❝❛❧ ♣②r❛♠✐❞s✳ ✻ Pr♦❜❧❡♠ ✷✳✺✳ ❆ ❝✉❜❡ ✐s ❝✉t ✐♥t♦ t❡tr❛❤❡❞r♦♥s✳ ❆t ❧❡❛st ❤♦✇ ♠❛♥② t❡tr❛❤❡❞r♦♥s ❞♦ ✇❡ ❣❡t❄ ❙♦❧✉t✐♦♥✿ ❊❛❝❤ ❢❛❝❡ ♦❢ t❤❡ ❝✉❜❡ ✐s ❛ sq✉❛r❡✱ s♦ ✐t s❤♦✉❧❞ ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦ ❛t ❧❡❛st t✇♦ ♣❛rts✳ ❙❡❧❡❝t t✇♦ ♦♣♣♦s✐♥❣ ❢❛❝❡s ✭♦❢ t❤❡ ❝✉❜❡✮✳ ❚❤❡r❡ ❛r❡ ❢♦✉r tr✐❛♥❣❧❡s ✇✐t❤♦✉t t✇♦ ❜❡✐♥❣ ✐♥ t❤❡ s❛♠❡ t❡tr❛❤❡❞r♦♥✳ ❚❤❡ t❡tr❛❤❡❞r❛❧ ❢❛❝❡s ❛r❡ ♦♥❡ ♦❢ t❤❡s❡ ❢♦✉r tr✐❛♥❣❧❡s ❛♥❞ t❤❡② ✇✐❧❧ ❤❛✈❡ ❛ t♦t❛❧ ✈♦❧✉♠❡ ♦❢ ♥♦ ♠♦r❡ t❤❛♥ ✱ ■♥ ❝❛s❡ ✐t ❝❛♥ ❜❡ ❝✉t ✐♥t♦ t❡tr❛❤❡❞r♦♥s✿ s❡❧❡❝t ✈❡rt✐❝❡s ✇✐t❤♦✉t t✇♦ ❜❡✐♥❣ ♦♥ t❤❡ s❛♠❡ ❢❛❝❡ ♦❢ ✇❤✐❝❤ ✐♥❞✐❝❛t❡s t❤❛t ✇❡ ♥❡❡❞ ♠♦r❡ t❤❛♥ ❢♦✉r t❡tr❛❤❡❞r♦♥s✳ t❤❡ ❝✉❜❡✳ ❲❡ ❣❡t ❛ t❡tr❛❤❡❞r♦♥✳ ❲✐t❤ t❤❡ ❢♦✉r ❢❛❝❡s ♦❢ t❤❡ t❡tr❛❤❡❞r♦♥ ❜❡✐♥❣ ❥♦✐♥❡❞ ✇✐t❤ ❢♦✉r ❢r♦♠ t❤❡ ♦t❤❡r ❢❛❝❡ ♦❢ t❤❡ s✉r❢❛❝❡✱ ✇❡ ❣❡t t❡tr❛❤❡❞r♦♥s t❤❛t ❝♦♠♣❧❡t❡❧② ❝♦✈❡r t❤❡ ❝✉❜❡ ✭❋✐❣✉r❡✮✳ ❨♦✉ ❝❛♥ ❢✐♥❞ ♠♦r❡ ❡①❡r❝✐s❡s ✐♥ ❘❡❢❡r❡♥❝❡ ❬✷❪✱ ❬✸❪✳ ✸ ❈✉t t❤❡ ❝✉❜❡ ❜② ❛ ♣❧❛♥❡ ✸✳✶ P♦ss✐❜❧❡ ❝r♦ss s❡❝t✐♦♥ ♦❢ t❤❡ ❝✉❜❡ ✇✐t❤ ❛ ♣❧❛♥❡✳ ✼ Pr♦❜❧❡♠ ✸✳✶✳ ❈❛♥ t❤❡ ❝r♦ss s❡❝t✐♦♥ ♦❢ ❛ ❝✉❜❡ ❛♥❞ ❛ ♣❧❛♥❡ ❜❡✳ ✳ ✳ ❛✮ ✳ ✳ ✳ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣❧❡❄ ❜✮ ✳ ✳ ✳ ❛ ❤❡①❛❣♦♥❄ ❙♦❧✉t✐♦♥✿ ■❧❧✉str❛t✐♦♥✳ ❊①❡r❝✐s❡ ✸✳✶✳ ❋♦r t❤❡ ❝✉❜❡ ABCDEF GH ✱ t❤❡ ♣❧❛♥❡ ✭p✮ ❣♦❡s t❤r♦✉❣❤ ✭❢✐❣✉r❡✮✳ ❈♦♥str✉❝t t❤❡ ❝r♦ss s❡❝t✐♦♥ ❝r❡❛t❡❞ ❜② t❤❡ ❨♦✉ ❝❛♥ ❢✐♥❞ ♠♦r❡ ❡①❡r❝✐s❡s ✐♥ ❘❡❢❡r❡♥❝❡ ❬✹❪✳ ✽ p−♣❧❛♥❡ ♣♦✐♥ts ❛♥❞ t❤❡ ❝✉❜❡s✳ M, N ❛♥❞ L ✹ P❛✐♥t ❈♦❧♦r ❝✉❜❡s Pr♦❜❧❡♠ ✹✳✶✳ ❆t ❧❡❛st ❤♦✇ ♠❛♥② ❞✐❢❢❡r❡♥t ❝♦❧♦rs ❝❛♥ ✇❡ ♣❛✐♥t t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ ❝✉❜❡ s♦ t❤❛t ❝♦♥♥❡❝t❡❞ ✈❡rt✐❝❡s ✭t❤♦s❡ ❧②✐♥❣ ❛❧♦♥❣ t❤❡ s❛♠❡ ❡❞❣❡✮ ❤❛✈❡ ❞✐❢❢❡r❡♥t ❝♦❧♦rs❄ ❙♦❧✉t✐♦♥✿ ❲❡ ♥❡❡❞ ❛t ❧❡❛st ❝♦❧♦rs✳ Pr♦❜❧❡♠ ✹✳✷✳ ❆t ❧❡❛st ❤♦✇ ♠❛♥② ❝♦❧♦rs ❝❛♥ ②♦✉ ♣❛✐♥t t❤❡ ❡❞❣❡s ♦❢ a ❝✉❜❡ s♦ t❤❛t ❡❞❣❡s s❤❛r✐♥❣ t❤❡ s❛♠❡ ✈❡rt❡① ❤❛✈❡ ❞✐❢❢❡r❡♥t ❝♦❧♦rs❄ ❙♦❧✉t✐♦♥✿ ❲❡ ♥❡❡❞ ❝♦❧♦rs✳ Pr♦❜❧❡♠ ✹✳✸✳ ❆t ❧❡❛st ❤♦✇ ♠❛♥② ❝♦❧♦rs ❞♦ ②♦✉ ♥❡❡❞ t♦ ♣❛✐♥t t❤❡ ❢❛❝❡s ♦❢ ❛ ❝✉❜❡ s♦ t❤❛t t❤❡ ❢❛❝❡s ♦♥ t❤❡ s❛♠❡ ❡❞❣❡ ❤❛✈❡ ❞✐❢❢❡r❡♥t ❝♦❧♦rs❄ ❙♦❧✉t✐♦♥✿ ❲❡ ♥❡❡❞ ❛t ❧❡❛st ❝♦❧♦rs✳ Pr♦❜❧❡♠ ✹✳✹✳ ■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ✇❡ ♣❛✐♥t t❤❡ ❢❛❝❡s ♦❢ ❛ ❝✉❜❡ ✐♥ ❜❧❛❝❦ ❛♥❞ ✇❤✐t❡ s♦ t❤❛t ❡❛❝❤ ❢❛❝❡ ♦♥❧② ❤❛s ♦♥❡ ❝♦❧♦r❄ ❈✉❜❡s t❤❛t ❝❛♥ ❜❡ r♦t❛t❡❞ t♦ ♦✈❡r❧❛♣ ❞♦ ♥♦t ❝♦✉♥t ❛s ❞✐❢❢❡r❡♥t✳ ✾ ❙♦❧✉t✐♦♥✿ ❲✐t❤ t✇♦ ❞✐❢❢❡r❡♥t ❝♦❧♦rs✱ t❤❡r❡ ❛r❡ ✇❛②s t♦ ♣❛✐♥t t❤❡ ❝✉❜❡✳ ■❢ ✇❡ ❛❧❧♦✇ ✉♥✐❝♦❧♦r ❝✉❜❡s✱ t❤❡r❡ ❛r❡ t✇♦ ♠♦r❡ ✇❛②s✱ ✇❤✐❝❤ ♠❛❦❡s ✐t ❛ t♦t❛❧ ♦❢ 10 s♦❧✉t✐♦♥s ❢♦r ❛ ♠❛①✐♠✉♠ ♦❢ t✇♦ ❝♦❧♦rs✳ ▲❡t ✉s ❤❛✈❡ ❛ ❧♦♦❦ ❛t t❤❡ ❞✐❢❢❡r❡♥t ✇❛②s ❛♥❞ ♣✉t t❤❡♠ ✐♥t♦ ❛ t❛❜❧❡✳ ❖❜✈✐♦✉s❧②✱ t❤❡r❡ ✐s ♦♥❧② ❝✉❜❡ ✇❤✐❝❤ ❤❛s 0, 1, ♦r r❡❞ ❢❛❝❡s✳ ■❢ t✇♦ ❢❛❝❡s ❛r❡ r❡❞✱ t❤♦s❡ t✇♦ ❢❛❝❡s ❝❛♥ ❜❡ ❡✐t❤❡r ❛❞❥❛❝❡♥t ♦r ♦♣♣♦s✐t❡ t♦ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡s❡ ❛r❡ t✇♦ ♦♣t✐♦♥s ❛♥❞ ♥❛t✉r❛❧❧② t❤❡r❡ ❛r❡ t✇♦ s✐♠✐❧❛r ♦♣t✐♦♥s ✇✐t❤ r❡❞ ❛♥❞ ❜❧✉❡ ❢❛❝❡s✳ ■♥ ❝❛s❡ ✇❡ ❤❛✈❡ t❤r❡❡ r❡❞ ❢❛❝❡s✱ ❧❡t ✉s ❛❞❞ ♦♥❡ r❡❞ ❢❛❝❡ t♦ t❤❡ ♣r❡✈✐♦✉s s❝❡♥❛r✐♦✳ ■❢ t❤❡ t✇♦ r❡❞ ❢❛❝❡s ❛r❡ ♦♣♣♦s✐t❡ ❡❛❝❤ ♦t❤❡r✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✇❛② ♦❢ ♣❛✐♥t✐♥❣ t❤❡ t❤✐r❞ ❢❛❝❡ r❡❞✳ ■❢ t❤❡ t✇♦ r❡❞ ❢❛❝❡s ❛r❡ ❛❞❥❛❝❡♥t✱ t❤❡♥ t❤❡r❡ ✐s ♦♥❡ ♠♦r❡ ✇❛② ♦❢ ♣❛✐♥t✐♥❣ t❤❡ ❝✉❜❡✱ ✇✐t❤ ❡❛❝❤ r❡❞ ❢❛❝❡ ❥♦✐♥✐♥❣ t❤❡ t✇♦ ♦t❤❡r r❡❞ ❢❛❝❡s ♦♥ t✇♦ s✐❞❡s✳ ❚❤✐s ❛r❡ ❛❧s♦ t✇♦ ♦♣t✐♦♥s✳ ◆✉♠❜❡r ♦❢ ❜❧❛❝❦ s✐❞❡s ◆✉♠❜❡r ♦❢ ✇❤✐t❡ s✐❞❡s ◆✉♠❜❡r ♦❢ ✇❛② ❆ t♦t❛❧ ♦❢ 10 3 2 1 ✇❛②s✳ Pr♦❜❧❡♠ ✹✳✺✳ ■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ✇❡ ♣❛✐♥t t❤❡ ❢❛❝❡s ♦❢ ❛ ❝✉❜❡ ✐♥ ❞✐❢❢❡r❡♥t ❝♦❧♦rs s♦ t❤❛t ❡❛❝❤ ❢❛❝❡ ❤❛s ♦♥❧② ♦♥❡ ❝♦❧♦r❄ ❈✉❜❡s t❤❛t ❝❛♥ ❜❡ r♦t❛t❡❞ t♦ ♦✈❡r❧❛♣ ❞♦ ♥♦t ❝♦✉♥t ❛s ❞✐❢❢❡r❡♥t✳ ❙♦❧✉t✐♦♥✿ ❚❤❡r❡ ❛r❡ 30 ✇❛②s✳ ❋✐rst ✇❡ ♣❛✐♥t ♦♥❡ ❢❛❝❡ r❡❞ ❛♥❞ ♣❧❛❝❡ t❤❡ ❝✉❜❡ st❛♥❞✐♥❣ ♦♥ t❤✐s ❢❛❝❡ ✭t❤✐s ❢❛❝❡ ✇✐❧❧ ❜❡ ❝♦✈❡r❡❞✮✳ ❚❤✐s ❧❡❛✈❡s ✺ ❢❛❝❡s✳ ❚❤❡r❡ ❛r❡ s♦ t❤❡r❡ ❛r❡ 4·3·2· =6 ✇❛②s t♦ ♣❛✐♥t t❤❡ t♦♣ ❢❛❝❡✳ ❚❤❡ ♦t❤❡r ❢♦✉r ❢❛❝❡s ❝❛♥ ❜❡ r♦t❛t❡✱ ✇❛②s✳ ❚❤✐s ♠❛❦❡s ❛ t♦t❛❧ ♦❢ · = 30 ✇❛②s✳ Pr♦❜❧❡♠ ✹✳✻✳ ■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ✇❡ ♣❛✐♥t t❤❡ ❢❛❝❡s ♦❢ ❛ ❝✉❜❡ ✐♥ ❝♦❧♦rs s♦ t❤❛t ❡❛❝❤ ❢❛❝❡ ❤❛s ♦♥❧② ♦♥❡ ❝♦❧♦r❄ ❈✉❜❡s t❤❛t ❝❛♥ ❜❡ r♦t❛t❡❞ t♦ ♦✈❡r❧❛♣ ❞♦ ♥♦t ❝♦✉♥t ❛s ❞✐❢❢❡r❡♥t✳ 75 ✇❛②s✳ ❡①❛❝t❧② ❢❛❝❡s ♦❢ t❤❡ s❛♠❡ ❝♦❧♦r✳ ■❢ t❤❡② ❛r❡ ♦♣♣♦s✐t❡ ❢❛❝❡s✱ t❤❡ ♦t❤❡r ❢❛❝❡s ❝❛♥ 4! ✇❛②s✳ ❍♦✇❡✈❡r✱ t❤❡ ❢❛❝❡s ♦❢ s❛♠❡ ❝♦❧♦r ❝❛♥ ❛❧s♦ ❜❡ s✇❛♣♣❡❞ ❜② r♦t❛t✐♦♥✱ ✇❤✐❝❤ ❙♦❧✉t✐♦♥✿ ❚❤❡r❡ ❛r❡ ❚❤❡r❡ ❝❛♥ ❜❡ ❜❡ ♣❛✐♥t❡❞ ✐♥ ❧❡❛✈❡s =3 ❞✐❢❢❡r❡♥t ❝♦❧♦r ❝♦♠❜✐♥❛t✐♦♥s✳ ❢❛❝❡s ❝❛♥ ❜❡ 24 = 12 ✇❛②s✱ ✇❤✐❝❤ ■❢ t❤❡ t✇♦ ❢❛❝❡s ♦❢ t❤❡ s❛♠❡ ❝♦❧♦r s❤❛r❡ ❛ ❝♦♠♠♦♥ ❡❞❣❡✱ t❤❡♥ t❤❡ r❡♠❛✐♥✐♥❣ ♣❛✐♥t❡❞ ✐♥ 4! = 24 ❞✐❢❢❡r❡♥t ✇❛②s✱ ❜✉t t❤❡s❡ ❝♦♥t❛✐♥ ❞♦✉❜❧❡s✳ ❚❤✐s ❧❡❛✈❡s ❜r✐♥❣s ✐t t♦ ❛ t♦t❛❧ ♦❢ 12 + = 15 ❞✐❢❢❡r❡♥t ✇❛②s✳ ▲❡t ✉s ♥♦t ❢♦r❣❡t t♦ ♠✉❧t✐♣❧② t❤✐s ♥✉♠❜❡r ✇✐t❤ ✭5 ❞✐❢❢❡r❡♥t ❝♦❧♦rs✮ t♦ ❣❡t × 15 = 75 ❞✐❢❢❡r❡♥t ✇❛②s✳ ❊①❡r❝✐s❡ ✹✳✶✳ ■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ✇❡ ♣❛✐♥t t❤❡ ❢❛❝❡s ♦❢ ❛ ❝✉❜❡ ✐♥ ❝♦❧♦rs s♦ t❤❛t ❡❛❝❤ ❢❛❝❡ ❤❛s ♦♥❧② ♦♥❡ ❝♦❧♦r❄ ❈✉❜❡s t❤❛t ❝❛♥ ❜❡ r♦t❛t❡❞ t♦ ♦✈❡r❧❛♣ ❞♦ ♥♦t ❝♦✉♥t ❛s ❞✐❢❢❡r❡♥t✳ ❆♥s✇❡r✿ ❚❤❡r❡ ❛r❡ 30 ✇❛②s✳ ❊①❡r❝✐s❡ ✹✳✷✳ ■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ✇❡ ♣❛✐♥t t❤❡ ❢❛❝❡s ♦❢ ❛ ❝✉❜❡ ✐♥ ❝♦❧♦rs s♦ t❤❛t ❡❛❝❤ ❢❛❝❡ ❤❛s ♦♥❧② ♦♥❡ ❝♦❧♦r❄ ❈✉❜❡s t❤❛t ❝❛♥ ❜❡ r♦t❛t❡❞ t♦ ♦✈❡r❧❛♣ ❞♦ ♥♦t ❝♦✉♥t ❛s ❞✐❢❢❡r❡♥t✳ ✶✵ ❆♥s✇❡r✿ ❚❤❡r❡ ❛r❡ 68 ✇❛②s✳ ❊①❡r❝✐s❡ ✹✳✸✳ ■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ✇❡ ♣❛✐♥t t❤❡ ❢❛❝❡s ♦❢ ❛ ❝✉❜❡ ✐♥ ♥ ❞✐❢❢❡r❡♥t ❝♦❧♦rs s♦ t❤❛t ❡❛❝❤ ❢❛❝❡ ❤❛s ♦♥❧② ♦♥❡ ❝♦❧♦r❄ ❈✉❜❡s t❤❛t ❝❛♥ ❜❡ r♦t❛t❡❞ t♦ ♦✈❡r❧❛♣ ❞♦ ♥♦t ❝♦✉♥t ❛s ❞✐❢❢❡r❡♥t✳ ❆♥s✇❡r✿ × (n6 + · n4 + 12 · n3 + · n2 )✳ 24 Pr♦❜❧❡♠ ✹✳✼✳ ■♥ ❤♦✇ ♠❛♥② ✇❛②s ❝❛♥ ✇❡ ♣❛✐♥t t❤❡ ❢❛❝❡s ♦❢ ❛ ❝✉❜❡ ✐♥ t❤r❡❡ ❝♦❧♦rs ✭r❡❞✱ ❜❧✉❡✱ ②❡❧❧♦✇✮ s♦ t❤❛t t❤❡r❡ ❛r❡ t✇♦ ❢❛❝❡s ♦❢ ❜❧✉❡✱ t✇♦ ♦❢ r❡❞✱ ❛♥❞ t✇♦ ♦❢ ②❡❧❧♦✇❄ ❈✉❜❡s t❤❛t ❝❛♥ ❜❡ r♦t❛t❡❞ t♦ ♦✈❡r❧❛♣ ❞♦ ♥♦t ❝♦✉♥t ❛s ❞✐❢❢❡r❡♥t✳ ❙♦❧✉t✐♦♥✿ ❚❤❡r❡ ❛r❡ ✇❛②s✳ ❛✮ ■❢ t❤❡ t✇♦ r❡❞ ❢❛❝❡s ❛r❡ ♦♣♣♦s✐t❡✱ t❤❡♥ ❡✐t❤❡r t❤❡ ♦t❤❡r ♣❛✐rs ❛r❡ ♦♣♣♦s✐t❡ t♦♦ ✭♦♥❡ ✇❛②✮ ♦r t❤❡② ❛r❡ ❛❞❥❛❝❡♥t ✭♦♥❡ ✇❛②✮✳ ❚❤✐s ♠❛❦❡s ✐t 1+1=2 ✇❛②s✳ ❜✮ ❲❤❡♥ t❤❡ t✇♦ r❡❞ ❢❛❝❡s ❛r❡ ❛❞❥❛❝❡♥t ✭t❤❡② ❤❛✈❡ ❛ ❝♦♠♠♦♥ ❡❞❣❡✮✱ t❤❡r❡ ❛r❡ t✇♦ ♣♦ss✐❜✐❧✲ ✐t✐❡s✳ ❲❡ ❡✐t❤❡r ❤❛✈❡ ♦♥❡ ♣❛✐r ♦❢ ♦♣♣♦s✐t❡ s✐❞❡s ✇✐t❤ t❤❡ s❛♠❡ ❝♦❧♦r✱ ♦r ❛❧❧ ♦♣♣♦s✐♥❣ s✐❞❡s ❛r❡ ♦❢ ❞✐❢❢❡r❡♥t ❝♦❧♦r✳ ❚❤❡r❡ ❛r❡ ✐t 2+2=4 ✇❛②s ♦❢ ♣❛✐♥t✐♥❣ t❤❡ ❢❛❝❡s ✐♥ ❡❛❝❤ s❝❡♥❛r✐♦✱ ✇❤✐❝❤ ♠❛❦❡s ✇❛②s✳ ❚❤✉s✱ ✐♥ t♦t❛❧✱ t❤❡r❡ ❛r❡ 2+4=6 ❞✐❢❢❡r❡♥t ✇❛②s t♦ ♣❛✐♥t t❤❡ ❝✉❜❡✳ Pr♦❜❧❡♠ ✹✳✽✳ ❈❛♥ ✇❡ ♣❛✐♥t t❤❡ ❢❛❝❡s ♦❢ ❞✐❢❢❡r❡♥t 2×2×2 ❙♦❧✉t✐♦♥✿ ❆ s♠❛❧❧ ❝✉❜❡s ✐♥ t✇♦ ❝♦❧♦rs s♦ t❤❛t ✇❡ ❝❛♥ ❜✉✐❧❞ ❝✉❜❡s ❢r♦♠ t❤❡♠❄ 2×2×2 ❝✉❜❡ ❤❛s × = 24 ❡①t❡r♥❛❧ ❢❛❝❡s ❛♥❞ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ ✐♥t❡r♥❛❧ ❢❛❝❡s✳ ❋♦r t❤✐s r❡❛s♦♥ ♦♥❧② t❤❡ ♣❛✐♥t✐♥❣ s❡❡♥ ♦♥ t❤❡ ❢✐❣✉r❡ ❝❛♥ ✇♦r❦✳ ❚❤✐s ✐s s✉✐t❛❜❧❡ ❛s ✇❡ ❝❛♥ ❜✉✐❧❞ t❤❡ ②❡❧❧♦✇ ❝✉❜❡ ❜② t✉r♥✐♥❣ ❛r♦✉♥❞ t❤❡ s♠❛❧❧ ❝✉❜❡s✳ Pr♦❜❧❡♠ ✹✳✾✳ 27 s♠❛❧❧ ❝✉❜❡s × × ❝✉❜❡ ♦❢ ❛✮ ■❢ ✇❡ ♣❛✐♥t t❤❡ ❢❛❝❡s ♦❢ ❝♦❧♦r✱ ❝❛♥ ✇❡ ❜✉✐❧❞ ❛ ✐♥ t✇♦ ❝♦❧♦rs ✐♥ ❛♥② ✇❛② s♦ t❤❛t ❡❛❝❤ ❢❛❝❡ ✐s ♦♥❡ ✉♥✐❢♦r♠ ❝♦❧♦r❄ ❜✮ ❈❛♥ ✇❡ ❞♦ ✐t ✇✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t ❡❛❝❤ ❝✉❜❡ ❤❛s ❛♥ ❡q✉❛❧ ♥✉♠❜❡r ♦❢ ❢❛❝❡s ♦❢ ❡❛❝❤ ❝♦❧♦r❄ ❙♦❧✉t✐♦♥✿ ❛✮ ■♠♣♦ss✐❜❧❡✱ ♣❛✐♥t 3×3×3 13 ❝✉❜❡s r❡❞✱ 14 ❜❧✉❡✳ ❇♦t❤ ❣r♦✉♣s ✇✐❧❧ ❜❡ ♣r❡s❡♥t ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ ❝✉❜❡✳ ✶✶ ❜✮ ■♠♣♦ss✐❜❧❡✱ ✐♥ t❤✐s ❝❛s❡✱ t❤❡ t✉❜❡ ❝❛♥ ❜❡ ♣❛✐♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛② ✭t❤❡ t♦♣ ❢❛❝❡ t❤❛t ✐s ♥♦t ✈✐s✐❜❧❡ ✐s ②❡❧❧♦✇✮✳ ◆♦♥❡ ♦❢ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ ❝✉❜❡ ❛r❡ s✉rr♦✉♥❞❡❞ ❜② t❤❡ s❛♠❡ ❝♦❧♦r✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s ♥♦t ❛❧✇❛②s ♣♦ss✐❜❧❡ t♦ ❜✉✐❧❞ ❛ ❝✉❜❡ ♦❢ ✉♥✐❢♦r♠ ❝♦❧♦r✳ 27 s♠❛❧❧ ❝✉❜❡s✳ ❲❤❛t ✐s t❤❡ ❧❛r❣❡st ♥✉♠❜❡r ♦❢ ❢❛❝❡s t❤❛t ✇❡ ❝❛♥ ♣❛✐♥t ❜✉✐❧❞ ❛ × × ❝✉❜❡ ♦❢ ✉♥✐❢♦r♠ ❝♦❧♦r✳ ❊①❡r❝✐s❡ ✹✳✹✳ ❚❤❡r❡ ❛r❡ r❡❞ s♦ t❤❛t ✇❡ ❝❛♥♥♦t ❊①❡r❝✐s❡ ✹✳✺✳ ❛✮ ❚❤❡r❡ ✐s ❛ 3×3×3 ❝✉❜❡✳ ❱❡rt❡① ❆ ✐s r❡❞ ✭✇❤✐❧❡ ❛❧❧ t❤❡ ♦t❤❡rs ❛r❡ ✇❤✐t❡✮✳ ❚❛♠ ✇❛♥ts t♦ ♠♦✈❡ t❤❡ r❡❞ ❝✉❜❡ t♦ t❤❡ ♦♣♣♦s✐t❡ ✈❡rt❡① ♦❢ t❤❡ ❝✉❜❡✳ ❙❤❡ ❝❛♥ ♦♥❧② ❝❤❛♥❣❡ t❤❡ ❧❛②❡r ✭❤♦r✐③♦♥t❛❧ ♦r ✈❡rt✐❝❛❧✮ ❝♦♥t❛✐♥✐♥❣ t❤❡ r❡❞ ❝✉❜❡ ✇✐t❤ ❛ ♥❡✐❣❤❜♦✉r✐♥❣ ♦♥❡✳ ❈❛♠ ✇❛♥ts t♦ st♦♣ ❤❡r ❢r♦♠ r❡❛❝❤✐♥❣ ❤❡r ❣♦❛❧ ❛♥❞ ✉s❡s ♠❛❣✐❝ t♦ ❞❡❝✐❞❡ ❤♦✇ ♠❛♥② ❝❤❛♥❣❡s ❚❛♠ ✐s ❛❧❧♦✇❡❞ t♦ ♠❛❦❡ ✭❛♥②✇❤❡r❡ ❜❡t✇❡❡♥ − 10✮✳ ❈❛♥ ❚❛♠ ❞♦ ✐t❄ ❜✮ ■❢ ❚❛♠✬s ❣♦❛❧ ✐s t♦ ♠♦✈❡ t❤❡ r❡❞ ❝✉❜❡ t♦ ❛ ♥❡✐❣❤❜♦✉r✐♥❣ ♣♦s✐t✐♦♥✱ ❝❛♥ s❤❡ ❞♦ ✐t❄ Pr♦❜❧❡♠ ✹✳✶✵✳ ❈❛♥ ✇❡ ♣❛✐♥t ✐♥t♦ ❞✐❢❢❡r❡♥t 3×3×3 27 s♠❛❧❧ ❝✉❜❡s ✐♥ ❞✐❢❢❡r❡♥t ❝♦❧♦rs s♦ t❤❛t t❤❡② ❝❛♥ ❜❡ ❥♦✐♥❡❞ ❝✉❜❡s ♦❢ ✉♥✐❢♦r♠ ❝♦❧♦r❄ ❙♦❧✉t✐♦♥✿ ▲❡t t❤❡ t❤r❡❡ ❝♦❧♦rs ❜❡ ❜❧✉❡✱ ②❡❧❧♦✇ ❛♥❞ ❣r❡❡♥✳ ❋♦r ❛ ❧❛r❣❡ ❝✉❜❡ ✇❡ ♥❡❡❞ · = 54 ❜❧✉❡ ❢❛❝❡s✳ ❲❡ ♥❡❡❞ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ t✐❧❡s ❢♦r t❤❡ ♦t❤❡r t✇♦ ❝♦❧♦rs t♦♦✳ ❆❧t♦❣❡t❤❡r ✇❡ ♥❡❡❞ t♦ ♣❛✐♥t ❝✉❜❡s ✭6 · 27✮✱ 162 ❢❛❝❡s✳ ❚❤✐s ✐s ❡q✉❛❧ t♦ t❤❡ ♥✉♠❜❡r ♦❢ ❢❛❝❡s ♦♥ 27 s♠❛❧❧❡r t❤✉s t❤❡r❡ ♠✐❣❤t ❜❡ ❛ ✇❛② t♦ s♦❧✈❡ t❤✐s ❡①❡r❝✐s❡ ✐❢ ✇❡ ♣❛✐♥t ❡❛❝❤ ❛♥❞ ❡✈❡r② ❢❛❝❡✳ ▲❡t ✉s ❤❛✈❡ ❛ ❧♦♦❦ ❛t t❤❡ ❜❧✉❡ ❢❛❝❡s✳ ❲❡ ♥❡❡❞ ✈❡rt✐❝❡s✳ ❲❡ ❛❧s♦ ♥❡❡❞ 12 ❝✉❜❡s t❤❛t ❤❛✈❡ ❜❧✉❡ ❢❛❝❡s t♦ s❡r✈❡ ❛s t❤❡ ❝✉❜❡s ✇✐t❤ t✇♦ ❜❧✉❡ ❢❛❝❡s t♦ ♣❧❛❝❡ ❜❡t✇❡❡♥ t❤❡ ✈❡rt✐❝❡s ♦♥ t❤❡ ❡❞❣❡s✱ ❛♥❞ ✻ ❝✉❜❡s t❤❛t ❤❛✈❡ ♦♥❡ ❜❧✉❡ ❢❛❝❡ t♦ ❣♦ ✐♥ t❤❡ ♠✐❞❞❧❡ ♦❢ ❡❛❝❤ ❧❛r❣❡r ❢❛❝❡✳ ✶✷ ▲❡t ✉s ❞❡♥♦t❡ t❤❡s❡ ✇✐t❤ B3 , B2 , B1 ✳ ❲❡ ♥❡❡❞ t❤❡ s❛♠❡ ❢r♦♠ t❤❡ ♦t❤❡r t✇♦ ❝♦❧♦rs✱ ❧❡t ✉s ❞❡♥♦t❡ t❤❡♠ s✐♠✐❧❛r❧②✳ ❚❤✉s✱ ✇❡ ♥❡❡❞ B3 , Y3 , G3 ❀ 12 B2 , Y2 , G2 ❛♥❞ B1 , Y1 , G1 ✳ ❚❤✐s ❝❛♥ ❜❡ ♠❛❞❡ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✿ ❝✉❜❡ ♦❢ B3 , Y3 ❀ ❝✉❜❡ ♦❢ Y3 , G3 ❀ ❝✉❜❡ ♦❢ G3 , B3 ❀ B2 , Y2 , G2 ❀ ❝✉❜❡s ♦❢ B3 , Y2 , G1 ❀ ❝✉❜❡s ❝✉❜❡s ♦❢ ♦❢ Y3 , G2 , B1 ❀ ❝✉❜❡s ♦❢ G3 , B2 , Y1 ✳ ❲❡ ❛❧s♦ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ✇❤❡t❤❡r t❤❡s❡ ❝✉❜❡s ❝❛♥ ❛❝t✉❛❧❧② ♣❛✐♥t❡❞ t❤✐s ✇❛② ❛♥❞ ✇❤❡t❤❡r t❤❡② ❝❛♥ ❜❡ r♦t❛t❡❞ ✐♥ t❤❡ ❝♦rr❡❝t ❝♦♥❢✐❣✉r❛t✐♦♥✱ ❜✉t ✇❡ ❧❡❛✈❡ t❤✐s ✉♣ t♦ t❤❡ ❘❡❛❞❡r✳ ❨♦✉ ❝❛♥ ❢✐♥❞ ♠♦r❡ ❡①❡r❝✐s❡s ✐♥ ❘❡❢❡r❡♥❝❡ ❬✸❪✱ ❬✺❪✳ ✺ P❛t❤s ✐♥ t❤❡ ❝✉❜❡ Pr♦❜❧❡♠ ✺✳✶✳ ❖♥ ❛ × × ❝✉❜❡✱ ❤♦✇ ♠❛♥② ♣❛t❤s ❣♦ ❢r♦♠ ♣♦✐♥t A t♦ ♣♦✐♥t B ✐♥ ❛ ✇❛② t❤❛t ♦♥❡ ❝❛♥ ♦♥❧② ♠♦✈❡ ❢r♦♠ ♥♦❞❡ t♦ ♥♦❞❡ ❛♥❞ ❝❛♥ ♦♥❧② ❣♦ ❡✐t❤❡r r✐❣❤t✱ ❧❡❢t ♦r ❞♦✇♥ ✭t❤❡r❡ ✐s ❛❧s♦ ❛ ♥♦❞❡ ❛t t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ❝✉❜❡✮❄ ❙♦❧✉t✐♦♥✿ ❚❤❡r❡ ❛r❡ ❛ t♦t❛❧ ♦❢ 90 ✇❛②s✳ ❚❤❡ ❝♦✉♥t✐♥❣ ♠❡t❤♦❞ ✐s ✐❧❧✉str❛t❡❞ ✐♥ t❤❡ ❢✐❣✉r❡✳ Pr♦❜❧❡♠ ✺✳✷✳ ❈♦♥s✐❞❡r ❛ ❝✉❜❡ ❢♦r♠❡❞ ❜② ❛ 3×3 ×3 ❣r✐❞✳ ❚❤❡ ✈❡rt✐❝❡s ❛r❡ ABCD ❛♥❞ EF GH ✇❤❡r❡ ✭A, E ✮❀ ✭B, F ✮❀ ✭C, G✮ ❛♥❞ ✭D, H ✮ ❛r❡ ♣❛✐rs ♦❢ ♦♣♣♦s✐♥❣ ✈❡rt✐❝❡s✳ ❙❤♦✇ t❤❛t ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❥♦✐♥ t❤❡ ❡❞❣❡s ♦❢ t❤❡ ❝✉❜❡ s♦ t❤❛t t❤❡ t❤r❡❡ ♣❛✐rs ♦❢ ❝♦♥t✐❣✉♦✉s ✈❡rt✐❝❡s ❤❛✈❡ ❞✐st✐♥❝t ♣❛t❤s ❧❡❛❞✐♥❣ t♦ ❡❛❝❤ ♦t❤❡r ✇✐t❤♦✉t ❤❛✈✐♥❣ ❛♥② ❝♦♠♠♦♥ ✈❡rt✐❝❡s✱ ❜✉t t❤❡r❡ ✐s ♥♦ s✉❝❤ ❝♦♥♥❡❝t✐♦♥ ❢♦r ♣❛✐rs ♦❢ ✈❡rt✐❝❡s✳ ●✉✐❞❡✿ ❘❡❢❡r t♦ t❤❡ ❞r❛✇✐♥❣✳ ✶✸ ❊①❡r❝✐s❡ ✺✳✶✳ ❖r❛♥❣❡ ✐s ❛❧✇❛②s t❤✐♥❦✐♥❣ ❛❜♦✉t ♠❛t❤✳ ❙❤❡ ✐s ❣♦✐♥❣ ❢♦r ❛ str♦❧❧ ❛r♦✉♥❞ ❤✐s ❣❧❛ss ✈✐❧❧❛✳ ❚❤❡ ♣❛t❤s ❛r❡ str❛✐❣❤t ❧✐♥❡s t❤❛t ❝♦♥♥❡❝t t❤❡ ♥♦❞❡s✳ ❇❡❢♦r❡ ❣♦✐♥❣✱ ❖r❛♥❣❡ ✇❛♥ts t♦ ❞r❛✇ ❛ ♣❛t❤ s♦ t❤❡ s❤❡ ♦♥❧② ❤❛s t♦ ❣♦ t❤r♦✉❣❤ ❡❛❝❤ ♥♦❞❡ ♦♥❝❡✳ ❈❛♥ ❖r❛♥❣❡ ❞r❛✇ s✉❝❤ ❛ ♣❛t❤❄ P❧❡❛s❡ ❤❡❧♣ ❤❡r ❢✐♥❞ t❤❡ s♦❧✉t✐♦♥✦ ❨♦✉ ❝❛♥ ❢✐♥❞ ♠♦r❡ ❡①❡r❝✐s❡s ✐♥ ❘❡❢❡r❡♥❝❡ ❬✺❪✳ ✻ ❖t❤❡r ❡①❡r❝✐s❡s ❊①❡r❝✐s❡ ✻✳✶✳ ❆♥ ❛♥❞ ❇✐♥❤ ❛r❡ ♣❧❛②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❝❡ ❣❛♠❡✳ ❖♥ ❆♥✬s ❞✐❝❡✱ t❤❡r❡ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥✉♠❜❡rs✿ 4, 6, 10, 18, 20, 22✳ ❇✐♥❤✬s ❞✐❝❡ ❤❛s 3, 9, 13, 15, 17, 25✳ ❇♦t❤ ♣❧❛②❡rs ✇✐❧❧ r♦❧❧ t❤❡✐r ♦✇♥ ❞✐❝❡ ❛♥❞ t❤❡ ✇✐♥♥❡r ✐s ✇❤♦❡✈❡r ❣❡ts t❤❡ ❧❛r❣❡r ♥✉♠❜❡r✳ ❲❤♦ ❤❛s t❤❡ ❛❞✈❛♥t❛❣❡ ✐♥ t❤✐s ❣❛♠❡❄ ❊①❡r❝✐s❡ ✻✳✷✳ ❆♥ ❛♥❞ ❇✐♥❤ ♣❧❛② ❛ ❞✐❝❡ ❣❛♠❡✳ ❇♦t❤ ❆♥ ❛♥❞ ❇✐♥❤✬s ❞✐❝❡ ❤❛✈❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ♦♥ t❤❡✐r ❢❛❝❡s✳ ❇♦t❤ ♣❧❛②❡rs r♦❧❧ t❤❡✐r ❞✐❝❡ ❛♥❞ t❤❡ ✇✐♥♥❡r ✐s ✇❤♦❡✈❡r ❣❡ts t❤❡ ❧❛r❣❡r ♥✉♠❜❡r✳ ■s ✐t tr✉❡ t❤❛t ✐❢ t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ s✐① ♥✉♠❜❡rs ♦♥ ❆♥✬s ❞✐❝❡ ✐s ❣r❡❛t❡r t❤❛♥ t❤❛t ♦♥ ❇✐♥✬s✱ t❤❡♥ ❆♥ ✇✐❧❧ ❤❛✈❡ ❛ ❜❡tt❡r ❝❤❛♥❝❡ ♦❢ ✇✐♥♥✐♥❣ t❤❛♥ ❇✐♥❤❄ ❊①❡r❝✐s❡ ✻✳✸✳ ❆♥ ❛♥❞ ❇✐♥❤ ♣❧❛② ❛ ❞✐❝❡ ❣❛♠❡✳ ❚❤❡r❡ ❛r❡ t❤r❡❡ ❡♠♣t② ❞✐❝❡ ♦♥ t❤❡ t❛❜❧❡✳ ❆♥ ✇r✐t❡s t❤❡ ♥✉♠❜❡rs ❢r♦♠ t♦ 18 ♦♥ t❤❡ ❞✐❝❡ ❛♥❞ ♣✐❝❦s ♦♥❡✳ ❇✐♥❤ ❝❤♦♦s❡s ♦♥❡ ♦❢ t❤❡ r❡♠❛✐♥✐♥❣ ❞✐❝❡✳ ❇♦t❤ ♣❧❛②❡rs r♦❧❧ t❤❡✐r ❞✐❝❡ ❛♥❞ t❤❡ ✇✐♥♥❡r ✐s ✇❤♦❡✈❡r ❣❡ts t❤❡ ❧❛r❣❡r ♥✉♠❜❡r ✭t❤❡ t❤✐r❞ ❞✐❝❡ ❞♦❡s ♥♦t ♣❧❛② ❛ r♦❧❡✳✮ ❲❤♦ ❤❛s t❤❡ ❛❞✈❛♥t❛❣❡✱ ❆♥ ♦r ❇✐♥❤❄ Pr♦❜❧❡♠ ✻✳✶✳ ❋r♦♠ ❛ 3×3 sq✉❛r❡ ❝✉t ♦✉t t❤❡ ❢r❛♠❡ ♦❢ ❛ ❝✉❜❡ ✐♥ ♦♥❡ ♣✐❡❝❡ s♦ t❤❛t t❤❡ ❝✉ts ❛r❡ ❡✐t❤❡r ♣❛r❛❧❧❡❧ ♦r ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❡❛❝❤ ♦t❤❡r✳ ❙♦❧✉t✐♦♥✿ ▲♦♦❦ ❛t t❤❡ ✐❧❧✉str❛t✐♦♥ ❢♦r t❤❡ ❝✉tt✐♥❣ s❡q✉❡♥❝❡✳ ▲♦♦❦✐♥❣ ❛t t❤❡ t♦♣ r♦✇✱ ✐t ♠✐❣❤t s❡❡♠ ❧✐❦❡ ✇❡ ❛r❡ ❝✉tt✐♥❣ ❛❧♦♥❣ t❤❡ ❧✐♥❡s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❣r✐❞✱ ✇❤✐❝❤ ✐s ❢❛❧s❡✳ ❊✈❡♥t✉❛❧❧②✱ ✇❡ ❝✉t t❤❡ ❢r❛♠❡ ♦❢ t❤❡ ❝✉❜❡ ♦✉t ♦❢ ❛ √ √ 2×2 sq✉❛r❡✳ ✶✹ ❲❡ ❝✉t ❛t ❛ 450 ❛♥❣❧❡ ❝♦♠♣❛r❡❞ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❣r✐❞✱ ❜✉t t❤❡ ❧✐♥❡s ♦❢ ❢♦❧❞✐♥❣ ❛r❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ❣r✐❞ ❧✐♥❡s✳ ✭❈♦❧❧❡❝t❡❞ ❛t ❍✉♥❣❛r✐❛♥ ❝❛♠♣s ✐♥ ❊①❡r❝✐s❡ ✻✳✹✳ ❆❧❧ 2005✮ ❡❞❣❡s ♦❢ ❛ ♣②r❛♠✐❞ ❛r❡ ❡q✉❛❧✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ✉s❡ ♦❢ s✉❝❤ ✐❞❡♥t✐❝❛❧ ♣②r❛♠✐❞s t♦ ❜✉✐❧❞ ❛ ❝✉❜❡ ✐♥ ❛ ✇❛② t❤❛t t❤❡✐r ✈❡rt✐❝❡s t❤❛t ❛r❡ ♣❛r❛❧❧❡❧ t♦ t❤❡✐r ❜❛s❡ t♦✉❝❤❄ ❊①❡r❝✐s❡ ✻✳✺✳ ❲❡ ❤❛✈❡ ❛ ❝✉❜✐❝ s❤❛♣❡❞ ❝❛❦❡ ✇✐t❤ ❛♥ ❡✈❡♥ ❝❤♦❝♦❧❛t❡ ❢r♦st✐♥❣ ♦♥ t❤❡ t♦♣ ❛♥❞ t❤❡ s✐❞❡s✳ ❍♦✇ ❞♦ ✇❡ ❝❛r✈❡ ✐t s♦ t❤❛t ❡✈❡r②❜♦❞② ❣❡ts t❤❡ s❛♠❡ ❛♠♦✉♥t ♦❢ ❝❛❦❡ ❛♥❞ ❝❤♦❝♦❧❛t❡ ❢r♦st✐♥❣❄ ❍♦✇ ❝❛♥ ✇❡ ❞♦ t❤✐s ✐❢ ✇❡ ❤❛✈❡ ❈❛♥ t❤❡ ❡①❡r❝✐s❡ ❜❡ s♦❧✈❡❞ ❢♦r n = 2, 3, 4, ♣❡♦♣❧❡❄ ❛♥② ♥❛t✉r❛❧ n ♥✉♠❜❡r❄ ❘❡❢❡r❡♥❝❡s ✶✳ ❆♥❞r❛s❢❛✐ ❇❡❧❛✿ ❱❡rs❡♥②♠❛t❡❦ ❣②❡r❡❦❡❦♥❡❦ ✳ ▼✉s③❛❦✐ ❦♦♥②✈❦✐❛❞â✳ ✷✳ ❘♦❦❛ ❙❛♥❞♦r✿ ✷✵✵✵ ❢❡❧❛❞❛t ❛③ ❡❧❡♠✐ ♠❛t❡♠❛t✐❦❛✐ ❦♦r❡❜♦❧✳ ❚②♣♦t❡① ❦✐❛❞♦✳ ✸✳ ❘♦❦❛ ❙❛♥❞♦r✿ ❙③❛❦❦♦r✐ ❢✉③❡t❡❦✱ ❆ ❦♦❝❦❛ ❚♦t❤ ❑♦♥②✈ ❦✐❛❞♦✳ ✹✳ ❙③❛t♠❛r✐ ❚✉♥❞❡✿ ■♥t❡r❛❦t✐✈✐t❛s ❛ ♠❛t❡♠❛t✐❦❛ ♦r❛♥✱ ❑✐✐♥❞✉❧♦♣♦♥t✉♥❦ ❛ ❦♦❝❦❛✳ ❙③❛❦❞♦❧❣♦③❛t✳ ❊▲❚❊✳ ✷✵✶✵✳ ✺✳ ❙③✉❝s ●❛❜♦r✿ ❆ ❦♦❝❦❛❦ ✈✐❧❛❣❛✳ ❑❛❧♠❛r ▲❛s③❧♦ ▼❛t❡❦✳✈❡rs❡♥② ♠♦❞s③❡rt❛♥✐ ❦✐❛❞✈❛♥②❛ ✷✵✶✻✳ ❍➔ ♥ë✐✱ ✶✺✴✵✸✴✷✵✶✽✳ ❊♠❛✐❧✿ ♥✈❧♦✐❅❤♦t♠❛✐❧✳❝♦♠✳ ✶✺