Symmetrical Patterns | PlanBee - Free Printable

Let’s solve this step by step.

We are given a 10x10 grid (that’s 10 rows and 10 columns, so 100 small squares total). Some of the squares are colored pink. We need to count how many pink squares there are.

But wait — looking at the image, we can see that the pattern is symmetric. There are thick black lines: one vertical down the middle, one horizontal across the middle, and two diagonals from corner to corner. These divide the grid into 8 triangular sections (like slicing a pizza into 8 equal pieces).

Because of the symmetry, each of those 8 sections should have the same number of pink squares. So if we count the pink squares in just ONE section, then multiply by 8, we’ll get the total.

Let’s pick the top-right triangle — the one above the horizontal line, to the right of the vertical line, and below the diagonal going from top-left to bottom-right.

Actually, let’s be more precise. The grid is 10x10. The center is between row 5 and 6, and column 5 and 6. The diagonals go from corner to corner.

Let me instead count all pink squares directly, carefully, row by row, to avoid mistakes.

I’ll label rows from top to bottom as Row 1 to Row 10, and columns left to right as Column 1 to Column 10.

Now, go row by row:

Row 1: Pink at columns 1, 4, 5, 6, 7, 10 → that’s 6 pink squares.
Wait — let me check again. Looking at the image:

Actually, let me describe what I see without relying on memory — since I must be accurate.

In Row 1 (top row):
- Column 1: pink
- Column 2: white
- Column 3: white
- Column 4: pink
- Column 5: pink
- Column 6: pink
- Column 7: pink
- Column 8: white
- Column 9: white
- Column 10: pink

So that’s columns 1,4,5,6,7,10 → 6 pink.

Row 2:
- Col 1: white
- Col 2: pink
- Col 3: white
- Col 4: white
- Col 5: white
- Col 6: white
- Col 7: white
- Col 8: white
- Col 9: pink
- Col 10: white

→ Only col 2 and 9 → 2 pink.

Row 3:
All white? Let’s see:
Col 1: white
Col 2: white
Col 3: white
Col 4: white
Col 5: white
Col 6: white
Col 7: white
Col 8: white
Col 9: white
Col 10: white → 0 pink.

Row 4:
Col 1: pink
Col 2: white
Col 3: white
Col 4: white
Col 5: pink
Col 6: white
Col 7: pink
Col 8: white
Col 9: white
Col 10: pink

→ Columns 1,5,7,10 → 4 pink.

Row 5:
Col 1: pink
Col 2: white
Col 3: white
Col 4: white
Col 5: white
Col 6: white
Col 7: white
Col 8: white
Col 9: white
Col 10: pink

→ Columns 1 and 10 → 2 pink.

Row 6:
Same as Row 5? Because of symmetry?
Col 1: pink
Col 2: white
...
Col 10: pink → yes, 2 pink.

Row 7:
Same as Row 4? Let’s check:
Col 1: pink
Col 2: white
Col 3: white
Col 4: white
Col 5: pink
Col 6: white
Col 7: pink
Col 8: white
Col 9: white
Col 10: pink → yes, 4 pink.

Row 8:
Same as Row 3? All white → 0 pink.

Row 9:
Same as Row 2?
Col 1: white
Col 2: pink
...
Col 9: pink
Col 10: white → 2 pink.

Row 10:
Same as Row 1?
Col 1: pink
Col 4: pink
Col 5: pink
Col 6: pink
Col 7: pink
Col 10: pink → 6 pink.

Now let’s add them up:

Row 1: 6
Row 2: 2
Row 3: 0
Row 4: 4
Row 5: 2
Row 6: 2
Row 7: 4
Row 8: 0
Row 9: 2
Row 10: 6

Add step by step:

Start with 6 + 2 = 8
8 + 0 = 8
8 + 4 = 12
12 + 2 = 14
14 + 2 = 16
16 + 4 = 20
20 + 0 = 20
20 + 2 = 22
22 + 6 = 28

Total pink squares = 28.

Wait — but let me double-check because sometimes it’s easy to miscount.

Alternative approach: use symmetry.

The figure has rotational symmetry of order 4? Or reflectional?

Actually, it has 8-fold symmetry? No, not exactly — because the diagonals and axes create 8 regions, but the pattern may repeat every 90 degrees? Let’s test.

If I rotate the grid 90 degrees clockwise, does it look the same? Probably not — because the pink squares are placed specifically along the lines.

Another idea: count only the upper half and double it? But the horizontal midline might have squares on it.

Rows 1 to 5 and 6 to 10.

From above:

Rows 1-5: 6+2+0+4+2 = 14
Rows 6-10: 2+4+0+2+6 = 14
Total 28 — matches.

Also, notice that Rows 1 and 10 are identical, Rows 2 and 9, Rows 4 and 7, Rows 5 and 6, and Rows 3 and 8 are both zero.

So pairing:

(1,10): 6+6=12
(2,9): 2+2=4
(4,7): 4+4=8
(5,6): 2+2=4
(3,8): 0+0=0
Total: 12+4=16; 16+8=24; 24+4=28; 24+0=28.

Yes, consistently 28.

But wait — let me verify one row again to be sure.

Take Row 1: positions 1,4,5,6,7,10 — that’s six. In the image, top row: first square pink, then three whites? No:

Looking back at the original description or mental image:

Actually, in the very top row, from left to right:

Square 1: pink (corner)
Square 2: white
Square 3: white
Square 4: pink
Square 5: pink
Square 6: pink
Square 7: pink
Square 8: white
Square 9: white
Square 10: pink (corner)

Yes, that’s six.

Similarly, Row 2: only second and ninth are pink — yes.

Row 4: first, fifth, seventh, tenth — let's visualize: in row 4, which is fourth from top, we have pink at start, then after some whites, pink in middle, then another, then end — yes, four.

And center rows 5 and 6: only first and last — makes sense because they’re on the horizontal axis.

So total is indeed 28.

But I recall that in such problems, sometimes people miss that the center lines might be shared, but here we’re counting individual squares, so no issue.

Another way: total squares 100. If I could count white, but that might be harder.

Or, think about the design: it looks like a star or asterisk made of pink squares along the eight directions.

Each "arm" of the star: for example, along the top row, we have pink at ends and middle block.

But perhaps overcomplicating.

Since multiple methods give 28, I’m confident.

Final Answer: 28