### Brain Teasers

# The Prime Tetrahedron

The six edges of tetrahedron ABCD measure 17, 23, 37, 67, 89 and 101 units. If the length of edge AB is 101, what is the length of edge CD?

### Hint

The sum of the lengths of two edges of a triangle must be greater than the length of the third edge.### Answer

Since AC+BC>AB and AD+BD>AB, the only combinations of AC+BC and AD+BD are 89+17 and 67+37, or 89+23 and 67+37.If they're 89+17 and 67+37, the remaining edge is 23, and we can make two triangles: 23, 67, 89 and 23, 17, 37.

If they're 89+23 and 67+37, the remaining edge is 17, but then there's no way to make the other two triangles.

Therefore, CD=23.

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