A Number Theory Approach to Problem Representation and Solution

A number t h e o r y approach to p rob lem s o l v i n g i s p r e s e n t e d u s i n g t h e tower o f cubes p u z z l e as an e x ample . Some c o n v e n t i o n a l methods o f s o l v i n g t he p u z z l e a re f i r s t d i s c u s s e d . The p u z z l e i s t h e n t h o r ough l y s t u d i e d u s i n g a n e n t i r e l y d i f f e r e n t app roach . S p e c i a l numbers a re used t o r e p r e s e n t t h e d i f f e r e n t c o l o r s on t h e s i d e s o f t he cubes . A d e s c r i p t i o n m a t r i x i s used t o compress t he p rob lem r e p r e s e n t a t i o n . The s o l u t i o n method becomes v e r y e f f i c i e n t because o f t h e conc i se r e p r e s e n t a t i o n o f t he p r o b l e m . Two t h e o rems a re deve loped to v e r i f y t h e s o l u t i o n method. A g e n e r a l i z a t i o n o f t he p u z z l e i s a l s o d i s c u s s e d . D e s c r i p t i v e Terms Number t h e o r y , p rob lem r e p r e s e n t a t i o n , p rob lem s o l u t i o n , p rob lem s o l v i n g , p u z z l e s , game p l a y i n g , p rob lem r e d u c t i o n e x i s t . A g e n e r a l i z a t i o n o f t he p u z z l e and some t h e o r e t i c a l background i s p r e s e n t e d i n S e c t i o n I V . A c o n c l u s i o n i s p r e s e n t e d i n S e c t i o n V . I I . Some S o l u t i o n Methods T h i s p u z z l e may be s o l v e d t h r o u g h the use o£ a s t a t e s p a c e r e p r e s e n t a t i o n 4 o f t h e p rob lem and by a p p l y i n g some sea rch t e c h n i q u e s to d i s c o v e r a s o l u t i o n . The f o l l o w i n g s u b s e c t i o n s d e s c r i b e some o f these methods . 1 . B l i n d Search : Cons ide r s t a c k i n g the cubes one by one. There a r e t w e n t y f o u r d i f f e r e n t ways t h a t a cube can be p o s i t i o n e d . Thus t h e s t a t e s p a c e r e p r e s e n t a t i o n o f t h e p rob lem may be e n v i s i o n e d as p i c t u r e d i n F i g u r e 1 . I . I n t r o d u c t i o n Two ma jo r s t eps i n p rob lem s o l v i n g a re t he p r o b lem r e p r e s e n t a t i o n and the p r o b l e m s o l u t i o n . These two s t e p s a r e v e r y much i n t e r r e l a t e d . Prob lem r e p r e s e n t a t i o n s have a g r e a t i n f l u e n c e upon t h e e f f i c i e n c y o f s o l u t i o n methods . A number t h e o r y 3 approach Is p roposed i n t h i s paper f o r b o t h t he r e p r e s e n t a t i o n and the s o l u t i o n o f p rob l ems . When a p p l i c a b l e , t h i s approach t u r n s o u t t o b e v e r y c o n c i s e i n t he r e p r e s e n t a t i o n and v e r y e f f i c i e n t i n t he s o l u t i o n o f p r o b lems. Even though n o t a l l p rob lems can be s o l v e d w i t h t h i s a p p r o a c h , i t s t i l l opens u p a d i r e c t i o n t h a t r e s e a r c h e r s may want t o l o o k i n t o t o d e v i s e b e t t e r methods i n s o l v i n g p r o b l e m s . The tower o f cubes p u z z l e i s used to i l l u s t r a t e t h i s approach I n p r o b l e m s o l v i n g . T o p l a y t h i s p u z z l e , a p l a y e r I s g i v e n f o u r cubes w i t h s i d e s i n f o u r d i f f e r e n t c o l o r s . Each cube may be d i f f e r e n t f r om the o t h e r In t he sense t h a t one cube may have t h r e e r e d s i d e s , one b l u e s i d e , one w h i t e s i d e , and one g reen s i d e , w h i l e ano the r cube may have two b l u e s i d e s , two g reen s i d e s , one red side, and one w h i t e s i d e . The f o u r cubes a re s h u f f l e d and t u r n e d randomly b e f o r e g i v e n t o t he p l a y e r . The o b j e c t i v e o f t he p l a y e r I s to s t a c k t he cubes into a tower so t h a t t h e r e are f o u r c o l o r s , a l l d i f f e r e n t , showing o n each s i d e o f t h e t o w e r . We can d e s c r i b e t h e s t a t e of a cube by a s i x t u p l e o f t h e fo rm < f r o n t , back , l e f t , r i g h t , t o p , bo t t am> . For examp le , t he t u p l e d e s c r i b i s a cube w i t h t h e f r o n t s i d e i n r e d , t h e back s i d e i n w h i t e , t h e l e f t s i d e i n b l u e , and s o f o r t h . W i t h any g i v e n i n p u t c o n f i g u r a t i o n o f t h e p u z z l e , t he computer can be used t o s o l v e t h e p u z z l e . D i f f e r e n t methods can be used and a r e b r i e f l y d i s c u s s e d i n S e c t i o n I I . I n S e c t i o n I I I , a method u s i n g t h e numb e r t h e o r y approach i s p roposed w h i c h w i l l f i n d a s o l u t i o n t o t h e p u z z l e e f f i c i e n t l y , o r i f a s o l u t i o n does n o t e x i s t , i n d i c a t e t h a t a s o l u t i o n does n o t F i g u r e 1 For f o u r cubes t h i s method can i n v o l v e a maximum o f 2 4 4 4 331,776 p a t h s . A l t h o u g h awkward, i t i s n o t p r o h i b i t i v e to use t h e b l i n d sea rch method on a mode r n compute r . However, i f one ex tends t h e game by a d d i n g a d d i t i o n a l cubes and c o l o r s , t h i s method r a p i d l y becomes u n f e a s i b l e .