Example:
Finite number of steps (own)
Finite number of steps (own)
Start with any real number other than $0$ and $1$. At each step, replace the number $x$ by either $1/x$ or $1x$. Prove that, if we can get from $a$ to $b$ in a finite number of steps, the maximum number of steps required is two.
Example:
Example:
"Everything should be made as simple as possible, but not simpler."  Albert Einstein
Re: Finite number of steps (own)
This problem is an interesting one. Though I think the maximum number of steps needed is 3.
"Je le vois, mais je ne le crois pas!"  Georg Ferdinand Ludwig Philipp Cantor

 Posts: 3
 Joined: Tue Dec 07, 2010 8:07 pm
 Location: Tokyo, Japan
Re: Finite number of steps (own)
Nice problem and nice solution.
I also think the number of steps needed is 3.
Let $f(x)=\dfrac{1}{x}$ and $g(x)=1x$.So $g\circ f= 1 \dfrac{1}{x},f\circ g=\dfrac{1}{1x},f\circ g\circ f=g\circ f\circ g=\dfrac{x}{x1}$.
And the fact that the set $\{id,f,g,g\circ f, f\circ g,f\circ g\circ f\}$ is closed under the composition of functions proves it all.
I also think the number of steps needed is 3.
Let $f(x)=\dfrac{1}{x}$ and $g(x)=1x$.So $g\circ f= 1 \dfrac{1}{x},f\circ g=\dfrac{1}{1x},f\circ g\circ f=g\circ f\circ g=\dfrac{x}{x1}$.
And the fact that the set $\{id,f,g,g\circ f, f\circ g,f\circ g\circ f\}$ is closed under the composition of functions proves it all.
"An equation for me has no meaning, unless it represents a thought of God." Srinivasa Iyengar Ramanujan
Re: Finite number of steps (own)
Yes, I'm sorry, the number of steps required should be 3. Nice solutions!
Try this slightly modified version I just posted: viewtopic.php?f=23&t=285
Try this slightly modified version I just posted: viewtopic.php?f=23&t=285
"Everything should be made as simple as possible, but not simpler."  Albert Einstein
Re: Finite number of steps (own)
সুন্দর সমস্যা এবং সমাধান। কিন্তু maximum এর জায়গায় minimum হবে মনে হইতেছে।