A store offers packages of $12$ pens for $\$10$ and packages of $20$ pens for $\$15$. Using only these two types of packages of pens, find the greatest number of pens $\$173$ can buy at this store.

[i]Proposed by James Lin[/i]
To solve this problem, we need to maximize the number of pens we can buy with $173 using the given packages. We have two types of packages:
- A package of 12 pens for $10.
- A package of 20 pens for $15.

We will use a systematic approach to determine the maximum number of pens we can buy.

1. **Define Variables and Constraints:**
   Let \( x \) be the number of 12-pen packages and \( y \) be the number of 20-pen packages. The total cost constraint is:
   \[
   10x + 15y \leq 173
   \]

2. **Express the Total Number of Pens:**
   The total number of pens \( P \) is given by:
   \[
   P = 12x + 20y
   \]

3. **Maximize the Number of Pens:**
   We need to maximize \( P \) under the given cost constraint. We will start by trying to use as many 20-pen packages as possible since they provide more pens per dollar.

4. **Calculate the Maximum Number of 20-Pen Packages:**
   The cost of one 20-pen package is $15. We calculate the maximum number of 20-pen packages we can buy with $173:
   \[
   y = \left\lfloor \frac{173}{15} \right\rfloor = \left\lfloor 11.5333 \right\rfloor = 11
   \]
   However, 11 packages cost:
   \[
   15 \times 11 = 165
   \]
   This leaves us with:
   \[
   173 - 165 = 8 \text{ dollars}
   \]
   Since 8 dollars is not enough to buy another 12-pen package (which costs $10), we need to check if we can get a better combination by reducing the number of 20-pen packages and adding 12-pen packages.

5. **Check Combinations:**
   - **10 packages of 20 pens:**
     \[
     10 \times 15 = 150 \text{ dollars}
     \]
     Remaining money:
     \[
     173 - 150 = 23 \text{ dollars}
     \]
     Number of 12-pen packages we can buy with 23 dollars:
     \[
     \left\lfloor \frac{23}{10} \right\rfloor = 2
     \]
     Total pens:
     \[
     10 \times 20 + 2 \times 12 = 200 + 24 = 224
     \]

   - **9 packages of 20 pens:**
     \[
     9 \times 15 = 135 \text{ dollars}
     \]
     Remaining money:
     \[
     173 - 135 = 38 \text{ dollars}
     \]
     Number of 12-pen packages we can buy with 38 dollars:
     \[
     \left\lfloor \frac{38}{10} \right\rfloor = 3
     \]
     Total pens:
     \[
     9 \times 20 + 3 \times 12 = 180 + 36 = 216
     \]

   - **8 packages of 20 pens:**
     \[
     8 \times 15 = 120 \text{ dollars}
     \]
     Remaining money:
     \[
     173 - 120 = 53 \text{ dollars}
     \]
     Number of 12-pen packages we can buy with 53 dollars:
     \[
     \left\lfloor \frac{53}{10} \right\rfloor = 5
     \]
     Total pens:
     \[
     8 \times 20 + 5 \times 12 = 160 + 60 = 220
     \]

   - **7 packages of 20 pens:**
     \[
     7 \times 15 = 105 \text{ dollars}
     \]
     Remaining money:
     \[
     173 - 105 = 68 \text{ dollars}
     \]
     Number of 12-pen packages we can buy with 68 dollars:
     \[
     \left\lfloor \frac{68}{10} \right\rfloor = 6
     \]
     Total pens:
     \[
     7 \times 20 + 6 \times 12 = 140 + 72 = 212
     \]

6. **Conclusion:**
   The maximum number of pens we can buy with $173 is achieved with 10 packages of 20 pens and 2 packages of 12 pens, giving us a total of 224 pens.

The final answer is \(\boxed{224}\).