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After his work with James Watson in discovering the double helical structure of DNA in the early 1950s, Francis Crick went on to be instrumental in determining the genetic code, proving that three-nucleotide groupings of RNA, called codons, coded for specific amino acids.

Even prior to experimentally proving that codons are three nucleotides in length, scientists were able to mathematically deduce that codons must be at least this long. With the knowledge that there are 20 amino acids found within biological proteins and four unique nucleotides in RNA, they could use simple math to determine minimal codon length.

For example, the number of possible different codons only one nucleotide long can be represented as ${4}^{1}$, or four codons. Since each amino acid requires its own unique codon, a genetic code based upon codons only one nucleotide long is insufficient to pair each amino acid to a unique codon.

Similarly, if each codon were two nucleotides in length, then the number of possible different codons that are two nucleotides long can be represented as ${4}^{2}$, or 16 codons, which is still several codons short of the 20 required to have one per amino acid. Thus, they were able to deduce that each codon had to be at least three nucleotides in length, such that the possible number of different codons was equal to ${4}^{3}$, or 64 codons.

Suppose NASA discovers life on Planet X. Scientists determine that nucleic acids on Planet X are composed of only two nucleotides, A and T. Additionally, proteins are composed of only 13 amino acids.

Assuming an otherwise similar mechanism for translation, what is the minimum number of nucleotides in each codon?

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