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I know the game is old, but I just finished it, and looking at existing walkthroughs and playthroughs I couldn't find any that full explain how the colour mixing works. Mostly they just give you the steps necessary to reach the solution. A lot of it is done just by trial and error, till you reach your answer.

In case anyone is interested, I compiled a full guide on how colour mixing in Lilly Looking Through works:

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RULES
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The game has three primary colours:
Red
Blue
Yellow

And three secondary colours:
Purple = Red + Blue
Orange = Red + Yellow
Green = Blue + Yellow

When you mix, you add a new colour on top of an existing colour. The result then is as follows:

• Adding Secondary on top of Secondary does nothing.

• Adding Primary on top of Primary does nothing.

• Adding Secondary on top of Primary:
a) Changes colour to Secondary, if Primary was one of its components. e.g. Purple onto Red gives Purple.
b) Otherwise, does nothing. e.g. Purple onto Yellow does nothing (stays Yellow).

• Adding Primary on top of Secondary:
a) Subtracts the Primary colour from Secondary, if it was one of its components. e.g. Blue onto Purple gives Red (because Purple - Blue = Red)
b) Otherwise, does nothing. e.g. Blue onto Orange does nothing (stays Orange).

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FULL LIST
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With those rules in mind, here are the possible colour changes:

Purple onto Red = Purple
Purple onto Blue = Purple
Orange onto Red = Orange
Orange onto Yellow = Orange
Green onto Blue = Green
Green onto Yellow = Green

Red onto Purple = Blue
Red onto Orange = Yellow
Blue onto Purple = Red
Blue onto Green = Yellow
Yellow onto Orange = Red
Yellow onto Green = Blue

For all other combinations: No change

==========

Once you know the rules, the actual puzzles become interesting. Especially the last one, where you are limited to only three starting colours.

I think the authors could have done a better job of familiarising the player with the rules. If they didn't want to state them explicitly, maybe they could have done it in a pictogram of some sort.
I think they tried to easy the player in, first by having a simple puzzle in the boat with only 2 primary colours (blue and yellow) and their secondary one (green). Then in the water the player had all six colours to play around with and understand the rules fully, then once that is done the final level has the actual puzzle where the player is restricted to only three colurs and two mixing spaces (tree and barrel) and must produce the desired output.
The colour feedback was also meant to explain it: primary colours would move away when subtracted, while secondary move forward.

Unfortunately it doesn't look like it was enough. Many players, myself included, just fumbled a bit till they got the solution and while they had a rough idea of how those colours work, they never got the full explanation.