CH. IV] GEOMETRICAL RECREATIONS 63 



similarly uses black counters or pieces or makes a nought on 

 each cell which he occupies. Whoever first gets three (or any 

 other assigned number) of his pieces in three adjacent cells 

 and in a straight line wins. There is no difficulty in giving 

 the complete analysis for boards of 9 cells and of 16 cells: but 

 it is lengthy and not particularly interesting. Most of these 

 games were known to the ancients*, and it is for that reason 

 I mention them here. 



Three-in-a-row. Extension. I may, however, add an elegant 

 but difficult extension which has not previously found its way, 

 so far as I am aware, into any book of mathematical recreations. 

 The problem is to place n counters on a plane so as to form as 

 many rows as possible, each of which shall contain three and 

 only three countersf. 



It is easy to arrange the counters in a number of rows 

 equal to the integral part of (n — l) 2 /8. This can be effected by 

 the following construction. Let P be any point on a cubic. 

 Let the tangent at P cut the curve again in Q. Let the tangent 

 at Q cut the curve in A. Let PA cut the curve in B, QB cut 

 it in C, PG cut it in D, QD cut it in E, and so on. Then the 

 counters must be placed at the points P, Q, A, B, .... Thus 9 

 counters can be placed in 8 such rows ; 10 counters in 10 rows ; 

 15 counters in 24 rows ; 81 counters in 800 rows ; and so on. 



If however the point P is a pluperfect point of the nth 

 order on the cubic, then Sylvester proved that the above 

 construction gives a number of rows equal to the integral 

 part of (n — 1) (n — 2)/6. Thus 9 counters can be arranged 

 in 9 rows; 10 counters in 12 rows; 15 counters in 30 rows; 

 and so on. 



These, however, are inferior limits and may be exceeded — 

 for instance, Sylvester stated that 9 counters can be placed 

 in 10 rows, each containing three counters; I do not know 

 how he placed them, but one way of so arranging them is 



* Beok de Fouquieres, Let Jeux des Anciens, second edition, Paris, 1873, 

 chap. xvih. 



+ Educational Times Reprints, 1868, vol. vm, p. 106 j Ibid. 1886, vol. xlv, 

 pp. 127-128. 



