Bung that fits Triangular, Square and Circular Holes
Problem
What shape of bung can be used to plug three different holes:
- one square
- one triangular
- one circular?
Solution
Let $r$ be the radius of the circle we want to plug.
Take a right circular cylindrical piece $B$ of the material for your bung whose base has radius $r$.
This will plug the circular hole.
Cut $B$ so that its height is $2 r$.
When you hold $B$ sideways, you see it will plug a square hole of side $2 r$.
Now draw a diameter $D$ of one of the bases of $B$.
Take two planes through $B$, both of which pass through $D$ such that they are both tangent to the other bases.
Slice away the material of $B$ on the other sides of the planes from the center of $B$.
Now $B$ has a triangular cross-section and it will plug a triangle whose sides are $\tuple {2 r, \sqrt 5 r, \sqrt 5 r}$.
$\blacksquare$
Historical Note
This puzzle is typical of those found in collections of Victorian amusements.
Sources
- 1633: Henry van Etten: Mathematicall Recreations (translated by William Oughtred from Récréations Mathématiques)
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Henry van Etten: $116$