## Finite number of steps (own)

For college and university level advanced Mathematics
nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### 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 \$1-x\$. 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:
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

Avik Roy
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Joined: Tue Dec 07, 2010 2:07 am

### 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)=1-x\$.So \$g\circ f= 1- \dfrac{1}{x},f\circ g=\dfrac{1}{1-x},f\circ g\circ f=g\circ f\circ g=\dfrac{x}{x-1}\$.
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

nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### 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
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

abir91
Posts: 52
Joined: Sun Dec 19, 2010 11:48 am

### Re: Finite number of steps (own)

সুন্দর সমস্যা এবং সমাধান। কিন্তু maximum এর জায়গায় minimum হবে মনে হইতেছে। Abir

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