Mathematical recreations and essays . A Bottom of Box B We may represent the various stages in the game by sup-posing that the blank space, occupying the sixteenth cell, ismoved over the board, ending finally where it started. The route pursued by the blank space may consist partly oftracks followed and again retraced, which have no efiect on thearrangement, and partly of closed paths travelled round, whichnecessarily are cyclical permutations of an odd number ofcounters. No other motion is possible. Now a cyclical permutation of n letters is equivalent ton—1 simple interchanges; accordingly a
Mathematical recreations and essays . A Bottom of Box B We may represent the various stages in the game by sup-posing that the blank space, occupying the sixteenth cell, ismoved over the board, ending finally where it started. The route pursued by the blank space may consist partly oftracks followed and again retraced, which have no efiect on thearrangement, and partly of closed paths travelled round, whichnecessarily are cyclical permutations of an odd number ofcounters. No other motion is possible. Now a cyclical permutation of n letters is equivalent ton—1 simple interchanges; accordingly an odd cyclical permu-tation is equivalent to an even number of simple , if we move the counters so as to bring the blank spaceback into the sixteenth cell, the new order must differ fi:omthe initial order by an even number of simple interchanges. Iftherefore the order we want to get can be obtained from thisinitial order only by an odd number of interchanges, theproblem is incapable of solution; if it can be obtai
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Keywords: ., bookcentury1900, bookdecade1920, booksubjec, booksubjectastrology
