The Islands of the Cartesian Sea — Mathcamp 2018 Puzzle Hunt

The ◣ and ◢ symbols in the Captain's Log record the sails used by Captain Cauchy to get between the islands (one sail per day). The colors represent different sails and the two versions of the triangle represent the two directions. Without loss of generality, say that ◣ is positive and ◢ is negative.

Each sail gives us a vector, and we can solve for the six vectors by combining the following observations:

Whenever Captain Cauchy returns to the origin, we get a nontrivial linear relation between the vectors.
From Day 16 to Day 20 to Day 22, Captain Cauchy travels from the origin to an island and then keeps going in the same direction for half the length. This can only happen with the islands at (-8,4) and (-12,6).
From Day 2 to Day 6, Captain Cauchy uses two blue sails and two yellow sails to get from the origin to an island, so both coordinates of that island must be even. The only remaining possibility for the island is (-4,4).
On Day 32, Captain Cauchy finds the treasure at unlucky coordinates, which refers to the island at (13,13).

Solving, we conclude that the red sail is (1,16), the orange sail is (2,9), the yellow sail is (3,-18), the green sail is (4,-1), the blue sail is (5,-20), and the purple sail is (6,5). These can be sorted in order of rainbow color, or by the first coordinate. The second coordinates spell out PIRATE after we convert from number to letter.