That's classical way to do genetics.
The solution is based on recombination and the chance of unique, double, triple, quadruple, etc, cross-over to occur. In that problem you'll have to assume that only double cross-over occurred. Triple and quadruple are very unlikely and single would destroy the chromosome.
So to start it up, you need to draw a map of all possible gene order (painful task) and look at all the possible location for cross-over and predict the possible recombinant and phenotype.
For example
Donor DNA___A______B______C_________D________E_____
Cross-over_1______2_____3_______4_________5_______6
Recipent____a_______b______c________d_________e______
Possible recombination ABCDE, aBCDE, Abcde, etc
Phenotype: doesn't need any compound, need compound a, need compound bcde, etc
Since all your plate have three amino acid added, you only have to look at recombination that exclude 3 genes / fix two genes.