How many triangles do you see in this image?

You thought you had the right answer, and yet... this famous puzzle is puzzling to even the most experienced eye! At first glance, you count a few triangles, then add them up as you progress through the “levels.” Simple? Not quite. The figure—a large triangle divided by three parallel lines at the base—hides more shapes than you can imagine. Ready to outsmart the trick and shine at your next quiz night? Follow the guide.

The Hidden Triangle Trap

Our brain loves to identify the most obvious small triangles and… forget the larger ones, formed by combining multiple areas. The result: we underestimate. In the classic configuration (4 rows formed by 3 lines parallel to the base), the correct answer is 27.  If
you got 26, you most likely missed a large compound triangle, often a triangle with the tip pointing upwards that extends over 3 levels, or a triangle “half-wide” on one side.

Step-by-step method: counting without mistakes

Identify the “upward” triangles by their height:

• Level height 1:10
• Height 2 levels: 6
• Height 3 levels: 3
• Height 4 levels: 1

Subtotal “up”: 10 + 6 + 3 + 1 = 20.

Add the “downward-facing” triangles resulting from the nesting of the segments: 7.

Total: 20 + 7 = 27.
Memory tip: think “20 ​​on top + 7 on the bottom” and you will avoid the classic forgetfulness.

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The magic formula explained simply

Looking for an elegant shortcut? For a large triangle divided into   n   segments arranged by lines parallel to the base (here   n   = 4), we can use the following formula:

T(n) = n × (n + 2) × (2n + 1) ÷ 8

Applied to   n   = 4:
T = 4 × 6 × 9 ÷ 8 = 27

This result   corresponds perfectly to our step-by-step counting. The beauty of this formula is that it “counts” all possible dimensions, including multilevel combinations. Ideal for testing your intuition... or for silencing a heated debate!

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Common Mistakes (and How to Avoid Them)

• Counting “by levels” without checking for large cross triangles: The typical culprit is a forgotten 3-level triangle.
• We omit the inverted (“downward”) triangles: they are less visible, but still present (7 in total).
• Add imaginary vertical lines: Some puzzle variations include real vertical lines, which change the total (but never to 26). Make sure you've set the position correctly before finishing.
• Double some triangles: follow a logical progression based on size to avoid duplicates.

Want to learn more?

• Change the number of rows: Try the formula with   n   = 3 or   n   = 5 and compare it to a manual count – perfect for training the eye.
• Time yourself: Give yourself 60 seconds to reach 27. The secret? Count the “up” part first, then add the “down” part.
• Tell a friend: teach a trick and you'll remember it forever (and win your next puzzle battle).

Because behind a simple triangle often hides a little lesson in logic... and the discreet pleasure of having the last word!

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