Hi everyone

 

Imagine a horizontal line of length 1. Divide it into 2 equal parts. Now pick a random side of the 2 equal parts: left, or right. If you pick the left half, divide it in 2 and now focus on the 'new' right half (of length 0,25). Divide that into 2 and now focus on the 'new' left half. Divide that into 2 and focus on the 'new' right side.

 

As you see, alternating between 'newly cut' left and right sides and divide that once again in 2.

 

I find that using this algorythm approaches 2 values: 1/3 for the left side, and 2/3 for the right side.

 

(1) Is this true and (2) how can this be proven?

 

Thanks.

 

F.

Edited June 24, 201510 yr by Function

if you imagine the line as length 1

the first cut is at 1/2

the second at 1/4

3/8, 5/16, 11/32...

 

that is to say

 

[latex]\frac{\frac{2^n-(-1)^n}{3}}{2^n}[/latex]

 

separate the 1/3

 

[latex]\frac{1}{3} \cdot \frac{2^n-(-1)^n}{2^n}[/latex]

 

The second term clearly gets damn close to one as n grows - so the whole expression tends to 1/3

if you imagine the line as length 1

the first cut is at 1/2

the second at 1/4

3/8, 5/16, 11/32...

 

that is to say

 

[latex]\frac{\frac{2^n-(-1)^n}{3}}{2^n}[/latex]

 

separate the 1/3

 

[latex]\frac{1}{3} \cdot \frac{2^n-(-1)^n}{2^n}[/latex]

 

The second term clearly gets damn close to one as n grows - so the whole expression tends to 1/3

 

Ah, thanks, imatfaal