Can you get them out of the prison?

Now that Anqi Li has taught us a lot about combinatorics, I think we should test our knowledge.

Consider 3 coins (green color) placed on an infinite chessboard (infinite in positive x and positive y direction) as shown. Each coin is a special coin in the sense that each coin can disintegrate into 2 new coins.Follow the diagram to understand this property.

The 4 squares that are encompassed by the red box are the prison. Using the above property, is it possible to have no coins inside the prison? Support your argument with a proof.

You can refer to the topics and brush up your concepts here

Zoom into the webpage using ctrl+scroll up to view the image clearly.

This problem has been adapted from the youtube channel Numberphile. You can view the problem here. Do not view the entire video as the solution is revealed.

#GameTheory

Easy Math Editor

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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}`	$\boxed{123}$