74 



GEOMETRICAL RECREATIONS 



[CH. IV 



In the diagram the square of the distance between W and F is 

 (32) a + (24) s ; hence the distance is 40 feet. 



W 



•s 



^. 



Problems with Counters placed in a row. Numerous 

 dynamical problems and puzzles may be illustrated with a 

 box of counters, especially if there are counters of two colours. 

 Of course coins or pawns or cards will serve equally well. 

 I proceed to enumerate a few of these played with counters 

 placed in a row. 



First Problem with Counters. The following problem must 

 be familiar to many of my readers. Ten counters (or coins) are 

 placed in a row. Any counter may be moved over two of those 

 adjacent to it on the counter next be} r ond them. It is required 

 to move the counters according to the above rule so that they 

 shall be arranged in five equidistant couples. 



If we denote the counters in their initial positions by the 

 numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, we proceed as follows. Put 

 7 on 10, then 5 on 2, then 3 on 8, then 1 on 4, and lastly* 9 on 

 6. Thus they are arranged in pairs on the places originally 

 occupied by the counters 2, 4, 6, 8, 10. 



Similarly by putting 4 on 1, then 6 on 9, then 8 on 3, then 

 10 on 7, and lastly 2 on 5, they are arranged in pairs on the 

 places originally occupied by the counters 1, 3, 5, 7, 9. 



If two superposed counters are reckoned as only one, 

 solutions analogous to those given above will be obtained by 



