The basic idea with this problem is that when translating, we do the following:
Look up word X in the row for language A, getting a list of meanings M1, M2, ...
Look up (B, M1), (B, M2), ..., corresponding to their translations Y1, Y2, ...
If a pivot C is inserted, then we do the following:
Look up word X in the row for language A, getting a list of meanings M1, M2, ...
Look up (C, M1), (C, M2), ..., corresponding to their translations Z1, Z2, ...
Then, for each translation Yi, look up Yi in the row for language C, getting a list of meanings M1', M2', ... (this is a superset of M1, M2, ... because Yi may have multiple meanings in C)
Look up (B, M1'), (B, M2'), ..., corresponding to their translations Y1', Y2', ...
This is how the pivot language allows us to reach more meanings in B.
D1 answers:
No pivot needed: M = { ocean }.
Pivot needed: M = { crow }, M' = { grass }. Need a language where "crow" and "grass" share a word, which is any of Bassa, Bété, Ejagham.
No pivot needed: M = { ocean }.
D2 answers:
a. M = { ocean, crow }, which map to "blue" and "black" in English respectively.
b. M = { ocean, crow }, Z = { tyre, ilyby }, M' = { ocean, crow, sky, grass, banana }, which map to "blue", "black", "green", and "yellow" in English.
c. M = { ocean, crow }, M' = { ocean, crow, sky, grass }. Need a language where "sky" and "grass" share a word with either "ocean" or "crow", but all others (plum, strawberry, banana, snow) are distinguished from "ocean" and "crow". This leaves Ejagham, Matsés, Seri. See problem C for visualization; we want the union of the "ocean" and "crow" components to only contain the additional colors "sky" and "grass".
As for D3, we need to figure out all the constraints, and then find the assignments that satisfy all of them.
M = { plum, strawberry }, M' = { crow }. Need a language where "crow" shares a word with "plum" and/or "strawberry": Bassa, Bété.
M = { plum, ocean }, M' = { sky }. Need a language where "sky" shares a word with "plum" and/or "ocean": Apinaye, Bassa, Bété, English, Teribe (only Tsafiki doesn't work).
M = { plum, strawberry }, M' = { plum }. This is a direct translation, so any pivot will do.
M = { plum, strawberry, banana }, M' = { grass, sky, ocean }. Need any language that doesn't distinguish one of "grass", "sky", or "ocean" from one of "plum", "strawberry", or "banana": Apinaye, Bassa, Bété, Teribe.
M = { banana }, M' = { snow }. Need a language where "snow" shares a word with "banana": only Bassa.
M = { grass }, M' = { plum, strawberry }. Need a language where "plum" and/or "strawberry" share a word with "grass": Apinaye, Bassa, Bété.
It happens that these constraints contain 1, 2, 3, 4, 5, 6 candidates, respectively! So we can satisfy them in the order of their restrictiveness: