Question

How many unique patterns are there in the image?

Hint:

To avoid confusion, scan the image row-by-row and cross out or mark duplicates systematically as you find them.

Answer 1

Correct Answer: 15

Step 1: Understanding the Question:
We are presented with a grid containing several complex star-shaped and mandala-like patterns.
Some of these patterns are identical duplicates of one another.
We need to count how many completely distinct (unique) pattern designs are present in the collection.

Step 2: Key Formula or Approach:
The approach involves a structured visual comparison:
1. Count the total number of icons on the board.
2. Group identical icons into pairs or sets.
3. Deduct duplicates to find the number of unique designs:
\[ \text{Unique Designs} = \text{Total Icons} - \text{Duplicates} \]

Step 3: Detailed Explanation:
1. Total Icon Count:
- Row 1: 7 patterns.
- Row 2: 6 patterns.
- Row 3: 6 patterns.
- Row 4: 5 patterns.
- Total icons = 24.
2. Identifying Duplicate Sets:
- Pair 1: Pattern 1 (Row 1, Column 1) is identical to Pattern 14 (Row 3, Column 1).
- Pair 2: Pattern 2 (Row 1, Column 2) is identical to Pattern 24 (Row 4, Column 5).
- Pair 3: Pattern 3 (Row 1, Column 3) is identical to Pattern 13 (Row 2, Column 6).
- Pair 4: Pattern 5 (Row 1, Column 5) is identical to Pattern 18 (Row 3, Column 5).
- Pair 5: Pattern 6 (Row 1, Column 6) is identical to Pattern 17 (Row 3, Column 4).
- Pair 6: Pattern 7 (Row 1, Column 7) is identical to Pattern 19 (Row 3, Column 6).
- Pair 7: Pattern 8 (Row 2, Column 1) is identical to Pattern 20 (Row 4, Column 1).
- Pair 8: Pattern 9 (Row 2, Column 2) is identical to Pattern 15 (Row 3, Column 2).
- Pair 9: Pattern 10 (Row 2, Column 3) is identical to Pattern 21 (Row 4, Column 2).
3. Counting Unique Patterns:
- There are 9 duplicate pairs, which accounts for $9 \times 2 = 18$ icons.
- The remaining $24 - 18 = 6$ icons are completely unique with no duplicates:
- Pattern 4 (Row 1, Column 4)
- Pattern 11 (Row 2, Column 4)
- Pattern 12 (Row 2, Column 5)
- Pattern 16 (Row 3, Column 3)
- Pattern 22 (Row 4, Column 3)
- Pattern 23 (Row 4, Column 4)
- Adding the 9 distinct designs from the pairs to the 6 unique individual designs gives:
\[ \text{Unique Patterns} = 9 + 6 = 15 \]

Step 4: Final Answer:
There are 15 unique patterns in the image.