A b s t r a c t . We consider the following generalization of the familiar '15-puzzle' which arises from issues in niemory manngrment, in distributed systems: I d e t G be a. p a p t i with n vertices w i t h k < n pebblrs nriniberet! 1 j...,k oil d i s t i n c t vrrticcs. A move c o i i s i a~s OF transf':':.rririg a pei'hlc to an a d j r~c (~i i t iinoccripicd vertex. Is one arraiigcnierit of th(= pebblcs r e ; d i a b l e lkrm a n o t h e r ? . We present c 1'-time decision algorithm, and provr inatching O(n") upper and lower !,ounds on the numbei of Inovrs required. These resalts extend thosc of Wilson ;1974), who considered G biconnrctcd and k=n-I, with n o analysis of number of moves.We nlbo consider the rlucsf ion of permutation group diameter. Driscoli and I h r s t (1983) ohtairic4 a polynoinial upper bound on thc diameter of groups grnrrnt,rd by bounded lrrigth cyclrs. We have tlie following ~u b e xponential bound for (crLain uriboondcd cy~les. 11' G'(on n lrtwrs) in gcnc-rntcd l)y cycles, one OF wliich !t,w prime Irngtii p , : 21ti3. a n d c' is prirnitivc, Lhcn G L A , or ,Sr, arid haq ti;arrid,er < P~'fit4n8.

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