82 GEOMETRICAL RECREATIONS [CH. IV 



points B lt ...B 6 . These latter points may be joined so as to form a 

 smaller regular re-entrant pentagon whose sides intersect in five points 

 C u ... 6 . The 16 points indicated are arranged as desired (The Canter- 

 bury Puzzles, 1907, p. 140). 



An arrangement of 18 counters in 9 rows, each containing 5 counters, 

 can be obtained thus. From one angle, A of an equilateral triangle 

 A A' A", draw lines AD, AE inside the triangle making any angles with 

 A A'. Draw from A' and A" lines similarly placed in regard to A' A" and 

 A" A. Let A'D' cut A"E" in F, and A'E' cut A"D" in <?. Then AFG is 

 a straight line. The 3 vertices of the triangle and the 15 points of inter- 

 section of AD, AE, AF, with the similar pencils of lines drawn from A', A", 

 will give an arrangement as required. 



An arrangement of 19 counters in 10 rows, each containing 5 counters, 

 can be obtained by placing counters at the 19 points of intersection of the 

 10 lines #= ±a, x= ±b, y= ±a, y= ±b, y= ±x: of these points two are 

 at infinity. 



Note. Page 69. The Great Northern Shunting Problem is effected thus, 

 (i) R pushes P into A. (ii) R returns, pushes Q up to P in A, couples 

 Q to P, draws them both out to F, and then pushes them to E. (iii) A 

 is now uncoupled, R takes Q back to A, and leaves it there, (iv) R returns 

 to P, takes P back to C, and leaves it there, (v) R running successively 

 through F, D, B comes to A, draws Q out, and leaves it at B. 



Note. Page 70. One solution of the Chifu-Chemulpo Puzzle is as 

 follows. Move successively wagons 2, 3, 4 up, i.e. on to the loop line. 

 [Then push 1 along the straight track close to 5 ; this is not a " move."] 

 Next, move 4 down, i.e. on to the straight track and push it along to 1. 

 Next, move 8 up, 3 down to the end of the track and keep it there tempo- 

 rarily, 6 up, 2 down, e down, 3 up, 7 up. [Then push 5 to the end of the 

 track and keep it there temporarily.] Next, move 7 down, 6 down, 2 up, 

 4 up. [Then push e along to 1.] Next, move 4 down to the end of the track 

 and keep it there temporarily, 2 down, 5 up, 3 down, 6 up, 7 up, 8 down 

 to the end of the track, e up, 5 down, 6 down, 7 down. In this solution 

 we moved e down to the track at one end, then shifted it along the track, 

 and finally moved it up to the loop from the other end of the track We 

 might equally well move e down to the track at one end, and finally 

 move it back to the loop from the same and. In this solution the pieces 

 successively moved are 2, 3, 4, 4, e, 8, 7, 3, 2, 6, 5, 5, 6, 3, 2, 7, 2, 5, 6, 3, 

 7, e, 8, 5, 6, 7. 



