# Calculational Puzzles

The puzzles on this page all require some sort of detailed mathematical calculation. This will often involve calculus or trigonometry. Most of these puzzles should be accessible to A-level maths students, though some may require slightly more advanced knowledge.

## 1. Intersecting Cylinders

What is the volume of the solid common to 3 intersecting (right-circular) cylinders of unit radius, whose axes are mutually perpendicular and meet at a common point?  For an easier problem, try finding the volume of the solid common to just two perpndicular intersecting cylinders. Can you see how this can be solved simply, without having to resort to calculus?

## 2. The best View

A statue of height l stands on a plinth. The top of the plinth is a height h above eve level. How far, d, should an observer stand from the base of the plinth in order to get the best view of the statue? You should find the value of d (in terms of h and l) which maximizes the angle, subtended by the statue from the point of observation.

[ You may assume that the statue and plinth are sufficiently thin that their widths may be neglected. ]

## 3. Circle in the Sky

What is the 3-space surface volume of a 4-dimensional unit hypersphere?

As an extension, can you find an explicit expression for the hypervolume of an n-dimensional unit hypersphere?

## 4. Shortest Route

In the perfectly flat district of Mathmoshire there are four towns, cunningly named: Alpha, Beta, Gamma, and Delta. They lie at the vertices of a large square, each side of which is 20km in length. The district planning department wishes to build a new road network connecting the towns. What is the minimum length of road which needs to be laid so it is possible to drive between every pair of towns just using the new roads? Where should the roads be laid?

## 5. Triangular Squares

Consider a plane tiled with identical squares, and let each vertex be a latice point. Is it possible to construct an equilateral triangle whose three corners all lie precisely on a lattice point? If so, then find it. If not, then find a proof of its impossibility.