Online Course Discussion Forum

MCIII Number Theory 2.6

 
 
Picture of Tina Jin
MCIII Number Theory 2.6
by Tina Jin - Thursday, 4 January 2024, 8:11 PM
 

Hello,

The solutions in the back of the book basically say the pattern is 2m for N=2^n+m. I test many numbers and of course they work, but I don't understand why.

Thanks,

Tina Jin

 
Picture of Dr. Kevin Wang
Re: MCIII Number Theory 2.6
by Dr. Kevin Wang - Saturday, 6 January 2024, 1:24 AM
 

You can start doing the operations until you have $2^n$ cards left in the stack.  How many steps have it been? Now that we have $2^n$ card, the bottom card will be the remaining one at the end.  So what is the current bottom card?

Picture of Tina Jin
Re: MCIII Number Theory 2.6
by Tina Jin - Sunday, 14 January 2024, 9:54 AM
 

I think I understand now. After m moves, the top 2m cards are now either discarded or below the (2^n+m)^th card. Only the even ones are below the (2^n+m)^th card, and the last one of those even cards (also the last card) would be 2m. The last card in a pile of 2^n cards will be the last card remaining since in a stack of cards with a power of 2 number of cards, it will be a skip the first, keep the second, skip the third, keep the fourth, ... skip the (2^n-1)^th, keep the (2^n)^th then repeat this process and only the (2^n)th card will always be kept. 

Thanks!
Tina Jin