234 MISCELLANEOUS PROBLEMS [CH. XI 



Gros asserted that it is possible to take from 64 to 

 80 steps a minute, which in my experience is a rather high 

 estimate. If we accept the lower of these numbers, it would 

 be possible to take off 10 rings in less than 8 minutes ; to take 

 off 25 rings would require more than 582 days, each of ten 

 hours work ; and to take off 60 rings would necessitate no less 

 than 768614,336404,564650 steps, and would require nearly 

 55000,000000 years work — assuming of course that no mistakes 

 were made. 



Initial position |° ° ° ° ° 10101 



After 1st step \ ° ° ° - ° 10110 \ 



„ 2nd ,, |° ° ° c 10111 f 



„ 3rd „ | ° ° 5— 5- 11000 



„ 4th „ |° c ° Q ° 11001) 



, 5th „ | ° p ° ° ° 11010 J 



» 6th , | ° ° ° H011 



„ 7th „ L°__. 2 moo 



loo o 



„ 8th „ fo o o m0 i 



- 9th - 1 ° I o° o ° miO) 



- 10th - 1 ° o o o o inn! 



Problems connected with a Pack of Cards. An ordinary 

 pack of playing cards can be used to illustrate many questions 

 depending on simple properties of numbers, or involving the 

 relative position of the cards. In problems of this kind, the 

 principle of solution generally consists in re-arranging the pack 

 in a particular manner so as to bring the card into some definite 

 position. Any such re-arrangement is a species of shuffling. 



I shall treat in succession of problems connected with 

 Shuffling a Pack, Arrangements by Rows and Columns, the 

 Determination of a Pair out of Jn(n + 1) Pairs, Gergonne's 

 Pile Problem, and the game known as the Mouse Trap. 



