递归问题——Concrete Mathematics

 

1.1汉诺塔问题

 

How to solve a recurrence problem

先猜后证明，采用数学归纳法来证明之

1）Look at small case.

2）Find and prove a mathematical expression for the quantity of interest

3）Find and prove a closed form for our mathematical expression

 

 

 

1.2 平面上的线lines in the plane

平面切割问题

Ln=L(n-1) +n

 

 

DEEP:"zig" instead of line

 

 

 

 

 

对于每一个zig, 实际上都失去2个rigion(2,3 ,as above)

 

所以Zn=L(2n)-2n

 

 

1.3 The Josephus problem

Determine the survivor's number:找到幸存者in  rounds

 

1. 偶数个人的情况， 2n people original . After one round, left is like starting out with n people, and each person's number has been doubled and decreased by 1:

J(2n)=2J(n)-1, for n>=1

1. 同1） 可得

J(2n+1)=2J(n)+1

This recurrence can make it possible to build a table to find patterns. 建立表格观察规律

Powers of 2 's magic!

 

 

Repeated application of J produces

Fixed Point : J(n)=n 可以得到当n=2v(n)-1时可以知道J(n)=n

 

General case (repertoire method清单法):