CH. VI] 



CHESS-BOARD RECREATIONS 



117 



in curved brackets, ( ) ; if it gives rise to only two solutions the 

 number which indicates it is placed in square brackets, [ ]; the 

 other fundamental solutions give rise to eight solutions each. 



On a board of 4" cells there is 1 fundamental solution : 

 namely, [3142]. 



On a board of 5 a cells there are 2 fundamental solutions : 

 namely, 14253, [25314]. It may be noted that the cyclic 

 solutions 14253, 25314, 31425, 42531, 53142 give five super- 

 posable arrangements by which five white queens, five black 

 queens, five red queens, five yellow queens, and five blue queens 

 can be put simultaneously on the board so that no queen can 

 be taken by any other queen of the same colour. 



On a board of 6 2 cells there is 1 fundamental solution: 

 namely, (246135). The four solutions are superposable. The 

 puzzle for this case is sold in the streets of London for a penny, 

 a small wooden board being ruled in the manner shown in 

 the diagram and having holes drilled in it at the points marked 

 by dots. The object is to put six pins into the holes so that 

 no two are connected by a straight line. 



On a board of 7 a cells there are 6 fundamental solutions : 

 namely, 1357246, 3572461, (5724613), 4613572, 3162574, 

 (2574136). It may be noted that the solution 1357246 gives 

 by cyclic permutations seven superposable arrangements. 



On a board of 8 2 cells there are 12 fundamental solutions : 

 namely, 25713864, 57138642, 71386425, 82417536, 68241753, 

 36824175, 64713528, 36814752, 36815724, 72418536, 26831475, 

 (64718253). The arrangement in this order is due to Mr Oram. 

 It will be noticed that the 10th, 11th, and 12th solutions some- 

 what resemble the 4th, 6th, and 7th respectively. The 6th 



