Circles in Triangles

The following pictures show n unit circles packed inside the smallest equilateral triangle (of side length s). Most of these have been proved optimal. When m = (n2+n)/2, s = 2(n-1) + 2√3. It is conjectured that one less circle does not change this.

 1. 2. 3. s = 2√3 = 3.464+Trivial. s = 2 + 2√3 = 5.464+Trivial. s = 2 + 2√3 = 5.464+Trivial.

 4. 5. 6. s = 4√3 = 6.928+Proved by Milano in 1987. s = 4 + 2√3 = 7.464+Proved by Milano in 1987. s = 4 + 2√3 = 7.464+Proved by Oler/Groemer in 1961.

 7. 8. 9. s = 2 + 4√3 = 8.928+Proved by Melissen in 1993. s = 2 + 2√3 + 2√33/3 = 9.293+Proved by Melissen in 1993. s = 6 + 2√3 = 9.464+Proved by Melissen in 1993.

 10. 11. 12. s = 6 + 2√3 = 9.464+Proved by Oler/Groemer in 1961. s = 4 + 2√3 + 4√6/3 = 10.730+Proved by Melissen in 1993. s = 4 + 4√3 = 10.928+Proved by Melissen in 1994.

 13. 14. 15. s = 4 + 2√6/3 + 10√3/3 = 11.406+Found by Melissen in 1993. s = 8 + 2√3 = 11.464+Found by Erdős/Oler in 1961. s = 8 + 2√3 = 11.464+Proved by Erdős/Groemer in 1961.

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