Problem 3

For each

k=1,2,3,4,5

, find necessary and sufficient conditions on

a>0

for the existence of a tetrahedron with

k

edges of length

a

and the remaining

6-k

edges of length

1

Step 3 of 5: Use the complementary four-edge case

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In plain words

Replacing every length by its reciprocal exchanges the roles of the $a$ -edges and unit edges.

0<a<\sqrt{2+\sqrt3}\quad(k=2),\qquad a>\frac1{\sqrt{2+\sqrt3}}=\sqrt{2-\sqrt3}\quad(k=4)

Detailed analysis

For $k=4$ , exchange the two edge lengths and rescale the $k=2$ result. The condition becomes $1/a<\sqrt{2+\sqrt3}$ , equivalently $a>1/\sqrt{2+\sqrt3}=\sqrt{2-\sqrt3}$ . Strict inequalities exclude the flat tetrahedra.