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In the last chapter of The Whispered World, one of the puzzles is to use a weird colour-mixing device to obtain a (royal) blue dye:

The colour-mixing puzzle

You can switch seven of the nine valves between plus and minus (the other two are stuck) and that’s it. It’s easy to obtain the solution by brute force, as there are only 2⁷ = 128 possible settings and you can heavily prune the combination tree by exploiting likely redundancies (e.g., when two different settings for the first mixing step yield the same result),

However, I want to actually understand this puzzle: What is the logic behind it? How can I solve it systematically?

All the walkthroughs I found either just give you the solution or instruct you to try around until you find the answer. In the creator’s commentary on this room, the Poki (the author of the puzzle) says that he is proud of the puzzle’s logic, but doesn’t give any hints.

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The Logic

Use the following diagram. Start in the centre. If the valve for an input colour is set to plus, go a step in the respective direction. If the valve is set to minus, go in the opposite direction. Do not make a step if the input colour is white. Do this for each input colour of a mixing step. The colour of the field you land on is the result of your mix:

Logic

There is no memory over successive mixing steps. The order of colours does not matter.

What the creators probably had in mind

You can also get the solution by applying the following rules:

  1. A valve set to minus returns the complementary colour: −red = green; −orange = blue; −yellow = violet
  2. Secondary colours decompose: green = blue + yellow; orange = red + yellow; violet = red + blue
  3. Three primary colours make white: red + blue + yellow = white
  4. Adding white to anything has no effect: X + white = X

Apply these rules until you end up with one of the following:

  • Only one kind of primary colour (red, blue, or yellow). In that case your mixing result is that colour.
  • Two kinds of primary colours. In that case you get their subtractive combination, no matter how many of each colour you have.

For example, you have:

  •    −blue + red − yellow
    = orange + red + violet
    = yellow + red + red + red + blue
    = red + red + white
    = red + red
    red

  •    −red + violet − yellow
    = green + violet + violet
    = blue + yellow + red + blue + red + blue
    = blue + white + red + blue
    = blue + red + blue
    violet

Digression: Why is this so difficult?

I take three issues with this puzzle:

  • The logic switches between additive and subtractive colour mixing: Rule 2 and 4 employ the logic of subtractive colour mixing, while Rule 3 is additive colour mixing.

  • Primary and secondary colours are treated differently. For example, as red and orange yields orange, one would expect −red and −orange to yield −orange, but it doesn’t:

    red + orange = red + red + yelloworange
    redorange = green + blue = blue + yellow + bluegreen (≠ blue = −orange)

  • You cannot evaluate in between. There is a reason why I use an arrow instead of an equals sign to describe the result of a mixing: The process is not symmetric. For example, since red + red yields red, you would expect that you can replace red + red with a single red in every occasion. However:

    red + red + yellow + blue = red
    red + yellow + blue = white

Solving the Puzzle

It’s best to think in reverse here:

Bottom

We have:

  • red (+) or green (−)
  • the result of the middle step which we cannot invert (because the valve is stuck).
  • violet (+) or yellow (−)

There are two options to obtain the desired blue under these constraints:

bottom

Thus, the middle step must either yield white or blue. Either way, the solution is −++.

Middle

We have:

  • violet (+) or yellow (−)
  • the result of the top step or its complement
  • orange (+) with a stuck valve

There is only one way to obtain blue or white under these constraints:

middle

Thus, the top step needs to yield green or red. In the first case, the solution is +++. In the second case, it’s +−+.

Top

We have:

  • blue (+) or orange (−)
  • yellow (+) or violet (−)
  • violet (+) or yellow (−)

There is only one way to obtain green (++−):

top

We cannot obtain red.

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