A Puzzle A Day: 2026-04-28

The ancient civilization of Xylos, fascinated by cosmic order, used precise geometric patterns in their sacred artifacts. One such artifact, 'The Stellar Hexagon', was constructed using exactly seven glowing crystals. These crystals were arranged on a flat surface to form a perfect, large hexagon: one crystal at the very center, and the remaining six crystals forming a regular hexagonal ring around it, all equidistant from the center and from their immediate neighbors in the ring. If an explorer is tasked with identifying all distinct equilateral triangles formed by connecting any three of these seven crystals, how many such triangles would they find?
Correct: 8
The 7 crystals form a classic 'honeycomb' pattern: one central crystal (let's call it C) and six outer crystals (P1-P6) arranged in a regular hexagon around it. All sides of the smaller equilateral triangles (like C-P1-P2) are of equal length 's', where 's' is the distance from the center to any outer crystal, and also the distance between any two adjacent outer crystals. 1. **Small Triangles (Side length 's'):** Each central crystal and two adjacent outer crystals form an equilateral triangle. These are C-P1-P2, C-P2-P3, C-P3-P4, C-P4-P5, C-P5-P6, and C-P6-P1. There are **6** such triangles. 2. **Large Triangles (Side length 's√3'):** By connecting alternating outer crystals, two larger equilateral triangles can be formed. These triangles connect crystals that are separated by one other crystal in the outer ring (e.g., P1, P3, P5). The distance between these alternating points is s√3. These two triangles are P1-P3-P5 and P2-P4-P6. There are **2** such triangles. All other combinations of three crystals will either form an isosceles triangle or a scalene triangle, but not an equilateral one. Therefore, the total number of distinct equilateral triangles is 6 (small) + 2 (large) = **8**.
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