[tromp]

Oops; I can't edit my post anymore. It went south starting from

> This reduces the equation to

We instead just ignore buttons f and g, and invert the equation

    A   1 0 1 1 0   a
    B   0 1 0 1 1   b
    C = 1 0 1 0 1   c
    D   1 1 0 1 1   d
    E   0 1 1 1 1   e

to

    a   0 1 0 1 0   A
    b   1 1 1 0 0   B
    c = 0 1 0 0 1   C
    d   1 0 0 1 1   D
    e   0 0 1 1 1   E

Showing which buttons to press in terms of exclusive-or of wrong lights, e.g. press button a if exactly one of lights B and D is wrong.