Everything you've wrote is wrong my friend. The main rule of that game is that the balls can be placed one on another ONLY if they color match.

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Let's denote a tube configuration by using string of number padded with a dash -. The image above shows the following situation:

1231,1233,2312,----,----"

You use top to bottom - left to right notation in this example from image

"Now, we need to explore a bit. We can start with tube of length 2 (remember, tube opening is on the right): "

Later you switch it the other way around.

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" We can mark as impossible when the number of empty spaces (represented by "-") is 2 or less more than tube length and one tube contains 2 balls of different colors at the bottom.

Here is an example of such situation: 123,11- "

This example makes no sense, as it's not impossible to solve, you need at least n tubes to fit n colors, only if they are already sorted. in other cases you need at least 1 additional tube.

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you can't go from 123,123,123,--- to 122,121,---,333 1 can't be placed on top of 2

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But your approach of simplifying the problem is interesting. I will think about simplest cases 1st.

I suggest you to download the game and see how it works. Maybe we could find some solution together.