When submitting problems, Steven the troll likes to submit silly names rather than his own. On day $1$, he gives no
name at all. Every day after that, he alternately adds $2$ words and $4$ words to his name. For example, on day $4$ he
submits an $8\text{-word}$ name. On day $n$ he submits the $44\text{-word name}$ “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute $n$.
1. On day 1, Steven gives no name at all, so the number of words is 0.
2. From day 2 onwards, he alternately adds 2 words and 4 words to his name. This means:
   - On day 2, he adds 2 words.
   - On day 3, he adds 4 words.
   - On day 4, he adds 2 words.
   - On day 5, he adds 4 words.
   - And so on.
3. We need to find the day \( n \) when the total number of words is 44.

Let's denote the number of words on day \( n \) as \( W(n) \). We can observe the pattern:
- On odd days (starting from day 1), the number of words added is 0, 4, 4, 4, ...
- On even days, the number of words added is 2, 2, 2, 2, ...

We can break this into two sequences:
- Sequence for even days: \( 2, 2, 2, \ldots \)
- Sequence for odd days: \( 0, 4, 4, 4, \ldots \)

4. Let's calculate the total number of words added up to day \( n \):
   - For even days: \( 2 \times \left\lfloor \frac{n}{2} \right\rfloor \)
   - For odd days: \( 4 \times \left\lfloor \frac{n-1}{2} \right\rfloor \)

Combining these, the total number of words \( W(n) \) is:
\[ W(n) = 2 \times \left\lfloor \frac{n}{2} \right\rfloor + 4 \times \left\lfloor \frac{n-1}{2} \right\rfloor \]

5. We need to find \( n \) such that \( W(n) = 44 \).

Let's solve this step-by-step:
- For \( n = 1 \): \( W(1) = 0 \)
- For \( n = 2 \): \( W(2) = 2 \)
- For \( n = 3 \): \( W(3) = 2 + 4 = 6 \)
- For \( n = 4 \): \( W(4) = 2 + 4 + 2 = 8 \)
- For \( n = 5 \): \( W(5) = 2 + 4 + 2 + 4 = 12 \)
- For \( n = 6 \): \( W(6) = 2 + 4 + 2 + 4 + 2 = 14 \)
- For \( n = 7 \): \( W(7) = 2 + 4 + 2 + 4 + 2 + 4 = 18 \)
- For \( n = 8 \): \( W(8) = 2 + 4 + 2 + 4 + 2 + 4 + 2 = 20 \)
- For \( n = 9 \): \( W(9) = 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 = 24 \)
- For \( n = 10 \): \( W(10) = 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 = 26 \)
- For \( n = 11 \): \( W(11) = 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 = 30 \)
- For \( n = 12 \): \( W(12) = 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 = 32 \)
- For \( n = 13 \): \( W(13) = 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 = 36 \)
- For \( n = 14 \): \( W(14) = 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 = 38 \)
- For \( n = 15 \): \( W(15) = 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 = 42 \)
- For \( n = 16 \): \( W(16) = 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 + 4 + 2 = 44 \)

Thus, \( n = 16 \) is the day when Steven submits a 44-word name.

The final answer is \( \boxed{16} \).