### Floored

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

### Getting an Angle

How can you make an angle of 60 degrees by folding a sheet of paper twice?

### Angle A

The three corners of a triangle are sitting on a circle. The angles are called Angle A, Angle B and Angle C. The dot in the middle of the circle shows the centre. The counter is measuring the size of Angle A in degrees. What is the smallest Angle A can be? What is the largest Angle A can be? What else do you notice about Angle A as you move the corners of the triangle around the circle?

# Pie Cuts

##### Stage: 3 Challenge Level:

The problem here is to cut a perfectly circular pie into n equal pieces using exactly three cuts. For what values of n is it possible to do this with straight cuts, and then how many different solutions are there? What if you are not restricted to straight cuts? There are some solutions here but also more challenges; it remains in the tough nut category!

Ling Xiang Ning, Allan, Tao Nan School, Singapore sent in this solution for 4 equal parts. It is not a trivial matter to find A but you might like to use an approximation method. Without loss of generality you can suppose the radius of the circle is 1 unit.

The problem is easier for 6 parts. Use 3 cuts along diameters to divide the pie into sectors with angles of 60 degrees at the centre. Find the endpoint by marking chords equal in length to the radius of the circle.

For 8 pieces Lewis O'Neill from Roundwood Primary School, Hertfordshire said "you cut it into 4, then you put it on its side and cut it collaterally". A super solution Lewis, well done.

Edwin Taylor remembered being set this in primary school and described the same solution as follows: "Looking from above the pie you cut it into quarters with two cuts. Then, looking at it side on, you make a horizontal cut parallel to the base of the pie. (I know the third cut isn't really a chord, but I assume that rule does not count if "cuts don't have to be straight line.)"

Here is another solution for 8 equal areas which gives equal shares of the icing on the top. Can you find the radius of the smaller circle?