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 5 of 5: Collect the necessary and sufficient conditions

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In plain words

The open endpoints record exactly when the tetrahedron flattens or two vertices coincide.

\boxed{k=1:\ 0<a<\sqrt3,\quad k=2:\ 0<a<\sqrt{2+\sqrt3},\quad k=3:\ a>0,\quad k=4:\ a>\sqrt{2-\sqrt3},\quad k=5:\ a>1/\sqrt3}

Detailed analysis

The hinge endpoints are degenerate and therefore excluded, while every interior value is attained continuously. Combining the cases gives the displayed necessary and sufficient conditions.