1. In DNA, the base pairing rules are:

   • $\mathrm{A}$ pairs with $\mathrm{T}$ by 2 hydrogen bonds.
   • $\mathrm{G}$ pairs with $\mathrm{C}$ by 3 hydrogen bonds.

2. Given one strand: $5'-\mathrm{G}-\mathrm{G}-\mathrm{C}-\mathrm{A}-\mathrm{A}-\mathrm{A}-\mathrm{T}-\mathrm{C}-\mathrm{G}-\mathrm{G}-\mathrm{C}-\mathrm{T}-\mathrm{A}-3'$

3. Count the number of each type of base in the given strand:

   • $\mathrm{G}$ appears at positions $1,2,9,10$ $\Rightarrow 4$
   • $\mathrm{C}$ appears at positions $3,8,11$ $\Rightarrow 3$
   • \mathrm[A} appears at positions $4,5,6,13$ $\Rightarrow 4$
   • $\mathrm{T}$ appears at positions $7,12$ $\Rightarrow 2$

4. Total number of $\mathrm{G}$ and $\mathrm{C}$ bases: $4+3=7$ Each $\mathrm{G-C}$ pair contributes $3$ hydrogen bonds. So hydrogen bonds from these pairs: $7 \times 3 = 21$

5. Total number of $\mathrm{A}$ and $\mathrm{T}$ bases: $4+2=6$ Each $\mathrm{A-T}$ pair contributes $2$ hydrogen bonds. So hydrogen bonds from these pairs: $6 \times 2 = 12$

6. Hence, total hydrogen bonds in the DNA double helix are: $21 + 12 = 33$

Therefore, the required integer answer is $33$.