ALL ABOUT TOYS, BUT A CHILD'S PLAY


Ken's father brought a toy for him. This toy was fitted 
with a wooden strip. There were three thick nails"A", "B", 
and "C" fitted to the strip. There were 35 plates made of 
brass on nail "A". these square plates were fitted in such 
a way that each plate was smaller in size than the one 
placed below it. The remaining nails "B" and"C" had no 
plates.

The plates on nail "A" were to be removed one by one and 
placed on nail "C" in such a way that the bigger plate 
should be below the smaller plate. There were some 
important guidelines and rules to be observed while 
removing the plates. They are as under: 

1.only one plate to be removed at a time.
2.nail "B" can be used while removing the plates.
3.at no time or place the bigger plates were to be placed 
over the smaller plates.

4. in any case, any of the three nails was to be used for 
keeping a plate.

Ken had removed 25 plates and began to play with 10 plates 
it look him one hour and ten minutes. In playing with the 
ten plates. As per the rules he removed the plates from 
nail "A" to nail "C".

Ken 's friend Sam  studied this toy and asked Ken if he 
could remove all the 35 plates from nail "A" to nail "C" 
and if so, in how much time.

After thinking for a moment, Ken replied that he could 
remove the plates within 7 to 8 hours.

Sam was not satisfied. He challenged Ken to do it within 
the stipulated time. Ken accepted the challenge. After some 
time Ken found out that it was not easy and Sam was right.

For this purpose he made a machine which speeded up the 
interchanging of plates 20 times.

Do you think Ken  will be able to transfer the plates from 
nail "A" to "C", with the help of this machine? What time 
do you think the machine will take to complete this?



SOLUTION-------- 
Ken would not be able to transfer the plates with the help 
of the machine. It is almost impossible. If the machine is 
kept on working for all 24 hours, the 35 plates may take 
81635.6 days within the stipulated conditions and rules. 

Note---- to transfer one plate, one move is involved, for 2 
discs, three moves; for 3, seven moves; for 4,15 moves are 
involved. And thus 35 plates are worked out on the 
following formula ; 2 to the power of 35--1, i.e. total 
34,35,97,38,367 moves. For 10 plates, in one minute 14.61 
moves are involved, whereas machine would do 20 times more 
work in one minute; i.e. it would make 292.28 moves. All 
this suggests that the task is well nigh impossible.