118 



CHESS-BOARD RECREATIONS 



[CH. VI 



solution is the only one in which no three queens are in a 

 straight line. It is impossible* to find eight superposable so- 

 lutions ; but we can in five typical ways pick out six solutions 

 which can be superposed, and to some of these it is possible to 

 add 2 sets of 7 queens, thus filling 62 out of the 64 cells with 

 6 sets of 8 queens and 2 sets of 7 queens, no one of which 

 can take another of the same set. Here is such a solution : 

 16837425, 27368514, 35714286, 41586372, 52473861, 68241753, 

 73625140, 04152637. Similar superposition problems can be 

 framed for boards of other sizes. 



For any reader who wishes to go further, I may add that it has 

 been shown that on a board of n? cells, there are 46 fundamental 

 solutions when n = 9, there are 92 when n = 10, there are 341 

 when n = ll, there are 1766 when ji=12, and there are 1346 

 when n=13. 



On any board empirical solutions may be found with but 

 little difficulty, and Mr Derrington has constructed the following 

 table of solutions : 



2.4.1.3 



for a board of 4* cells 



2.4.1.3.5 



2.4.6.1.3.5 



2.4.6.1.3.5.7 



2.4.6.8.3.1.7.5 



2.4.1.7.9.6.3.5.8 



2.4.6.8.10.1.3.5.7.9 



2.4.6.8.10.1.3.5.7.9.11 



2. 4. 6. 8. 10. 12. 1.3. 5. 7. 9. 11 



2.4.6.8.10.12.1.3.5.7.9.11.13 



9.7.5.3.1.13.11.6.4.2.14.12.10.8 



15 . 9 . 7 . 5 . 3 . 1 . 13 . 11 . 6 . 4 . 2 . 14 . 12 . 10 . 8 



2 . 4 . 6 . 8 . 10 . 12 . 14 . 16 . 1 . 3 . 5 . 7 . 9 . 11 . 13 . 15 



2.4.6.8.10.12.14.16.1.3.5.7.9.11.13.15.17 

 4.6.8.10.12.14.16.18.1.3.5.7.9.11 

 4 . 6 . 8 . 10 . 12 . 14 . 16 . 18 . 1 . 3 . 5 . 7 . 9 . 11 . 13 . 15 

 .10.8.6.4.2.20.18.16.14.9.7.5.3.1.19.17.15.13 



. 13 . 15 . 17 



.17.19 



2 



2. 



12 



21 . 12 . 10 . 8 . 6 . 4 . 2 . 20 . 18 . 16 . 14 . 9 . 7 . 5 . 3 . 1 . 19 . 17 . 15 



11 



5" 



6" 



7 2 



8 2 



9 2 



10 2 



11 s 



12 a 



13 2 



14» 



15 2 



16 a 



17 a 



18" 



19 2 



20 2 



, 212 



* See G. T. Bennett, The Messenger of Mathematics, Cambridge, June, 1909, 

 vol. xxxix, pp. 19—21. 



