Four corners of a square pursue each other clockwise, and after each unit step mark out the new smaller square. Show that the squares sizes in the sequence decrease in size to 1 in the limit.

# Author: Marshall W. Buck

## The hexagonal dart board

25 darts land inside a regular-hexagonal dart board with edge length the square root of 3. Show that at least five of the darts land inside the same unit circle.

## Wrapping presents

Six squares are wrapped together with red ribbons. Do they cross at a single point?

## Hidden circle II

What is the locus of orange points for which the purple lines are parallel? The fixed cevians of the triangle BCD are angle bisectors.

## Rolling level

Blue and green circles with tangents. The green tangent line goes through the blue tangent point and the points IJH are collinear. Show that the blue and orange lines are parallel.

## The Cheshire Cat grin

What is the total area of the green teeth relative to the large circle area? (Assume the inner circle radius is 80% of the outer. The 32 lines forming the tooth sides are concurrent, and spaced at equal angles apart.)

## Criss-crossing the bay

A boatsman races to cross Isosceles Bay five times, finishing at End. A bicyclist pedals around the bay. How many times faster must the bicyclist travel in order to be sure to beat the boat to End?

## Corner to corner

A rectangle inside a rectangle. What is the green area?

## All-seeing eye

If the radius shrinks by the factor 1/2 for each smaller circle, what fraction of the whole area is represented by all the blue crescents (separated by the red crescents)? Assume the circles keep going inward forever.

## Cycling of tangents

Four circles share one point. The quadrilateral has vertices which are intersections of pairs of circles, and three of its coloured edges are tangent to the circle of the same colour. Show that the fourth edge is also tangent to its circle.