Magic Hexagon - Free Printable

Let’s count the hexagons step by step.

We can count them row by row, from top to bottom.

Top row: There are 3 hexagons in a straight line at the very top.

Second row: Below that, there are 4 hexagons — they fit between the ones above.

Third row (middle): This is the widest part — there are 5 hexagons here.

Fourth row: Below the middle, it starts narrowing again — 4 hexagons.

Bottom row: At the very bottom, there are 3 hexagons.

Now let’s add them up:

3 (top)
+ 4 (second)
+ 5 (middle)
+ 4 (fourth)
+ 3 (bottom)
= 19

Wait — let me double-check by counting another way. Maybe group them differently.

Alternatively, we can think of this as a symmetric shape. Let’s label rows from top to bottom as Row 1 to Row 5.

Row 1: 3
Row 2: 4
Row 3: 5
Row 4: 4
Row 5: 3

Same numbers. 3+4=7; 7+5=12; 12+4=16; 16+3=19.

But wait — I’ve seen similar patterns before. Sometimes people miscount because some hexagons look like they’re shared or overlapping, but in this case, each hexagon is distinct and fully drawn.

Let me try counting visually in my mind again:

Start from the center — there’s one central hexagon.

Around it, how many? In a honeycomb pattern, each layer adds more.

Actually, maybe that’s overcomplicating.

Let me just go back to row-by-row — it’s reliable.

Row 1 (top): left, middle, right → 3

Row 2: under the gaps of row 1 → 4

Row 3: full width → 5

Row 4: same as row 2 → 4

Row 5: same as row 1 → 3

Total: 3 + 4 + 5 + 4 + 3 = 19

I think I made a mistake earlier when I thought it was 19 — but let me recount with actual positions.

Imagine labeling columns too.

Actually, let me draw it mentally with coordinates.

Or better — let’s use a different method: count all visible outlines.

Each hexagon has 6 sides, but they share sides, so that won’t help directly.

Best to stick with row counting.

Wait — I recall that in such a diamond-shaped hexagon grid, the total number is often calculated as n² for certain sizes, but here it’s not square.

Another idea: this looks like a “hexagonal number” pattern.

The centered hexagonal numbers are: 1, 7, 19, 37...

Oh! 19 is a centered hexagonal number.

That matches our count.

Centered hexagonal number formula: H_n = 3n(n-1) + 1

For n=3: 3*3*2 +1 = 18+1=19

Yes! So if this is the 3rd centered hexagonal number, it should have 19 hexagons.

And visually, it fits: one in center, then a ring of 6 around it (total 7), then another ring of 12 around that (7+12=19).

Perfect.

So whether we count row by row or use the mathematical pattern, we get 19.

Final Answer:
19