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Let's break down the problem step by step to solve for how many square counters there are.

### Information given:
1. There are three types of counters: **purple squares**, **purple triangles**, and **red triangles**.
2. **70% of the purple counters are triangles**.
3. **60% of the triangular counters are purple**.
4. There are **14 red counters**.
   
We need to find the number of **square counters**.

### Step 1: Define variables
Let:
- \( p_s \) = number of purple square counters.
- \( p_t \) = number of purple triangle counters.
- \( r_t \) = number of red triangle counters.

The total number of purple counters is \( p_s + p_t \).

### Step 2: Use the percentage information
- **70% of the purple counters are triangles**, so:
  \[
  p_t = 0.7 \times (p_s + p_t)
  \]
  This simplifies to:
  \[
  p_t = 0.7p_s + 0.7p_t
  \]
  Rearranging:
  \[
  0.3p_t = 0.7p_s
  \]
  Solving for \( p_t \):
  \[
  p_t = \frac{7}{3} p_s
  \]

### Step 3: Total triangles
There are **two types of triangles**: purple triangles (\( p_t \)) and red triangles (\( r_t = 14 \) from the problem).
- **60% of the triangles are purple**, so:
  \[
  p_t = 0.6 \times (p_t + r_t)
  \]
  Substituting \( r_t = 14 \):
  \[
  p_t = 0.6 \times (p_t + 14)
  \]
  This simplifies to:
  \[
  p_t = 0.6p_t + 8.4
  \]
  Rearranging:
  \[
  0.4p_t = 8.4
  \]
  Solving for \( p_t \):
  \[
  p_t = \frac{8.4}{0.4} = 21
  \]

### Step 4: Solve for \( p_s \)
Using \( p_t = \frac{7}{3} p_s \) and \( p_t = 21 \), we can solve for \( p_s \):
\[
21 = \frac{7}{3} p_s
\]
Multiplying both sides by 3:
\[
63 = 7p_s
\]
Solving for \( p_s \):
\[
p_s = \frac{63}{7} = 9
\]

### Final Answer:
There are **9 square counters**.

Would you like more details or have any questions? Here are some related questions to further explore this:

1. What if the percentage of purple counters being triangles was 60% instead of 70%?
2. How would the problem change if there were 20 red counters instead of 14?
3. What if 80% of triangular counters were purple instead of 60%?
4. How many total counters are in the bag, including squares and triangles?
5. Can we derive the ratio of purple counters to red counters?

**Tip:** Always check your work by re-substituting your final answers into the original conditions of the problem.

A bag contains three types of counters: purple squares, purple triangles, and red triangles. 70% of the purple counters are triangles. 60% of the triangular counters are purple. There are 14 red counters. How many square counters are there?

Learn how to calculate the number of square counters in a bag, given that 70% of the purple counters are triangles and 60% of triangular counters are purple. This problem uses algebra and percentages. Suitable for students in Grades 6-8.

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