Easter color mixing
Task number: 4503
At Acme Corporation they have developed food colours that change colour when exposed to heat. They plan to release a new product before Easter: Magic eggs. This is a pack of three eggs that initially look like normal, white eggs, but when cooked, they change colour and become coloured Easter eggs, each one a different colour.
For a long time, managers could not agree on what colours they wanted to sell the eggs in. Originally, they planned to produce these three colours:
- purple (25.0% cyan and 75.0% magenta)
- red (50.0% magenta and 50.0% yellow)
- blue (87.5% cyan and 12.5% yellow)
But at the last minute, they decided on these three colors:
- turquoise (77.5% cyan, 10.0% magenta and 12.5% yellow)
- green (40.0% cyan, 10.0% magenta and 50.0% yellow)
- pink (15.0% cyan and 85.0% magenta)
In the confusion caused by last-minute changes, staff of the Acme Corporation logistics centre made a crucial mistake: They sent (numbered) barrels with the original three colours to some production facilities, while to others they sent barrels with the new three colours. Easter is coming up, and they do not have time to send the new colors again. However, they cannot allow the packages from different facilities to contain eggs of different colours, because if the colors of the boiled eggs do not exactly match the promotional materials, they would be at risk of a false advertising lawsuit.
At the production plants they cannot detect which trio of colors they got, but they can mix their colors in any ratio they want. Determine what three ratios they are to mix their colors in at each plant, to produce three different coloured eggs in each plant, such that the three colours are the same across all plants.
Author: Benjamin Swart, 2024Solution
We were given two alternative bases of the CMY color space.
We would like to find vectors that have the same representation in both bases. So we construct a transition matrix and determine its eigenvalues. This matrix has 1 as one of its eigenvalues, so we find the corresponding eigenvector. This gives us the first ratio: we mix 40 % of the first barrel, 20 % of the second barrel, and 40 % of the third barrel to get 45 % cyan, 40 % magenta, and 15 % yellow.
The eigenvalue 1, however, has a multiplicity of only 1, and so this is the only vector with a sum of components of 1 that has the same representation in both bases. However, the transition matrix also has an eigenvalue \(-1\). This means that there is a pair of vectors that the transition matrix swaps. That is, the representation of one of them in the first basis corresponds to the representation of the other in the second basis. The assignment does not mention that all plants must mix the same colors in the same order, so such a pair together with the already found ratio satisfies the assignment.
It is easy to find such a pair: Determine the set of eigenvectors to the eigenvalue 1 of the square of the transition matrix. We choose any one of them different from the already found ratio, and the other one is easily calculated by multiplying it by the transition matrix.
One possible solution is to choose these two ratios:
- 20 % of the second barrel and 80 % of the third barrel
- 80 % of the first barrel and 20 % of the second barrel
Then, independently of the plant, we get the following second and third colours:
- 20 % cyan, 70 % magenta and 10 % yellow
- 70 % cyan, 10 % magenta and 20 % yellow
