Expert opinion: a clear answer… at the beginning
To settle the matter, several geometry specialists were consulted. All—or almost all—arrived at the same answer: 18 triangles. Their method? Start from the top vertex and observe how many pairs of points can form a base on each of the three horizontal lines. Three levels, six possibilities per level… voilà, 18. Easy peasy.
The explanation is elegant, logical, reassuring. You think to yourself, "Ah, finally! The mystery is solved!" Except it isn't. The story could have ended there, but the internet never sleeps…
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