Jubang

There are five each balls colored white or black in a row. By "rearrangement", five balls from right will be inserted between each balls in to left-side balls. For example ○ ○ ○ ○ ○ ● ● ● ● ● original → ○ ● ○ ● ○ ● ○ ● ○ ● n = 1

How many times of rearrangement need to be same as original alignment? Is there any efficient way to solve?

#NumberTheory

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

Looks like $\boxed{6}$ moves will return us to the original configuration.

Starting configuration:

$\circ\circ\circ\circ\circ\bullet\bullet\bullet\bullet~\bullet$

6 moves:

$\circ\bullet\circ\bullet\circ\bullet\circ\bullet\circ~\bullet$
$\circ\bullet\bullet\circ\circ\bullet\bullet\circ\circ~\bullet$
$\circ\bullet\bullet\bullet\bullet\circ\circ\circ\circ~\bullet$
$\circ\circ\bullet\circ\bullet\circ\bullet\circ\bullet~\bullet$
$\circ\circ\circ\bullet\bullet\circ\circ\bullet\bullet~\bullet$
$\circ\circ\circ\circ\circ\bullet\bullet\bullet\bullet~\bullet$

Obviously, but interestingly, the 2 circles on the extreme right and left will never change colour.

David Stiff - 3 months ago

Thank you so much!!What do you think my solving process showed above?

Tomoki Yoshii - 3 months ago

You're welcome!

Hmm...very interesting. There's quite a fascinating pattern going on behind the scenes!

David Stiff - 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$