The magic formula explained simply
Looking for an elegant shortcut? For a large triangle subdivided 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 perfectly matches our step-by-step count. The beauty of this formula is that it "counts" all possible sizes for you, including multi-level combinations. Ideal for checking your intuition... or silencing a heated debate!
Common mistakes (and how to avoid them)
- Counting “by levels” without checking the large cross-level triangles: the typical culprit is a forgotten 3-level triangle.
- Omit the inverted triangles (“downwards”): they are less visible, but still present (7 in total).
- Add imaginary vertical lines: some variations of the puzzle include real vertical lines, which changes the total (but never to 26). Make sure you have the exact configuration before concluding.
- Doubling some triangles: follow a logical progression by size to avoid duplicates.
Want to go further?
- Change the number of rows: test the formula with n = 3 or n = 5 and compare with a manual count — perfect for training your eye.
- Time yourself: give yourself 60 seconds to reach 27. The secret? Count the "upwards" first, then add the "downwards".
- Explain it to a friend: teaching a trick means remembering it permanently (and winning your next puzzle battle).
Because behind a simple triangle there often lies a little lesson in logic… and the discreet pleasure of having the last word!
