The 15 Puzzle

■Richard A. tor, says the London ïruth, has devoted his energies to the solution of the "Fiften," or "Boss" puzzle, and has published some three or four learned articles on tlie subject. He lias calculated that there are no fewer than 20,922,789,888 positions in which the puzzle may be placed, if an aspirant desires to attempt its solution. Of these he has satisfied himself that one-half ar soluble, and the remaining half insoluble. The soltóility of the problem depends upon a utrange law of relation between the line in which the vacant square occurs, and the ninnber of what Mr. Proctor terras " displacements." If, for instance, we read the box im the ordinary way, beginning with the upper horizontal line from left to right, aud flnd it to begin with 5, there are then four displacements, because 5 comes before 1, 2, 3 and 4, whereas it ought to follow them. If the next figure be 3, there are then two more displacements, because 3 comes before 1 and 2, whereas it ought to follow them. Now, according to Mr. Proctor, if the number of displacements are even, and the vacant, block is i one of the horizontal lines, reading from left to right - that is to say, in the scond line from the top or in the bottom line - the prolem is soluble. The same is the case if the number of displacement are uneven, and the vacant block be in au uneven line. But where the displacements are uneven, and the lina of the vacant block even or vice veraa, the problem is insoluble.