000RM.dvi

1010 Cycle decompositions

39.3 Dudeney’s Canterbury puzzle

Take nine counters numbered 1 to 9, and place them in a row

in the natural order. It is required in as few exchanges of pairs

as possible to convert this into a square number. As an ex-

ample in six moves we give the following: (7846932), which

give the number 139854276, which is the square of 11826.

But it can be done in much fewer moves.

The square of 12543 can be found in four moves, as follows.

123456789

(25) 153426789

(37) 157426389

(38) 157426839

(34) 157326849

The squares of 25572, 26733, and 27273 can also be obtained in four
moves.

• 123456789

653927184

•

123456789

714653289

•

123456789

743816529

(^1) Puzzle 128 of [Dudeney]