Toni's Tiara: Student's Notes
Introduction
These notes are designed to provide you with hints to help you while you work towards solving this problem and also to help you see a much larger picture of both the problem and of mathematics itself. The activity is available online at:
http://www2.nzmaths.co.nz/frames/brightsparks/tonisTiara.asp?applet
http://www2.nzmaths.co.nz/frames/brightsparks/tonisTiara.asp?applet
The notes below are split into four phases:
- Understanding and solving specific problems
- Generalising
- Proving
- Extending
The four phases are the same as those used by mathematicians when they are working on research problems. Of course, if you only want to go as far as solving the Toni’s Tiara problem for specific polygons, you won’t need to go through all of these phases. However, we hope that you will want to go further than this. The problem is certainly set up for you to find conjectures about all the possible polygons that can be achieved by slicing through a cube.
Below we have provided a number of hints for you to use if you need help. Try to do as much of the problem as you can yourself, and only use the hints if you get stuck.
Phase 1: Understanding and solving specific problems
At first glance this is a really very simple problem. All you have to do is to make a plane cut through a cube and see what shapes are formed on the exposed faces. The activity provides you with a red cube. The white outline on the cube is the shape you get on the exposed face by cutting the cube with a plane (not explicitly shown in the diagram). If you click and drag the cube itself you can make it rotate to expose other white shapes. If you move the arrow on the vertical scale on the left this is equivalent to moving the cube backwards and forwards through the cutting plane.
After some experimenting you might get some idea of how the faces might look. You can then tackle the Toni’s Tiara problem. You will have to show that you can actually make all the shapes that you say are possible in order to open the chest. So it’s a good idea to be sure what shapes can be made before tackling the retrieval of Toni’s tiara.
Hint 2
If you are not sure about what shapes can be made by cutting a cube then the first thing you should try is experimenting with the cube.
Hint 2
Make a list of the shapes you know how to make, because each time you try to unlock the chest you are given a different list of shapes to select from.
Hint 3
You should be able to find some shapes easily, but make sure that you make a complete list, as those you are challenged to make are described specifically (not just triangle – what kind of triangle?).
Hint 4
If you are asked to make a shape that you don’t recognize the name of you can quit and click on the question mark to access the glossary of shapes. This describes the features of all the shapes you may be asked to make. You can also use it to make a list of all the shapes you might be asked to try to make. (Note that the Glossary assumes that the cube you are working with has side lengths of 1 unit. You should also note that ‘√ x 1 rectangles’ should be ‘√2 x 1 rectangles’.)
Phase 2: Generalising
Hint 5
There are five types of triangle that you might be asked to make:
- Right angled
- Equilateral
- Scalene
- Obtuse angled
- Isosceles
Which of them are possible?
Hint 6
Three of the triangles are possible. The right angled triangle and the obtuse angled triangle are not possible.
Equilateral triangle 
Scalene triangle
Isoceles triangleHint 7
There are seven types of quadrilateral that you might be asked to make:
- Square
- 2x1 rectangle
- √2 x 1 rectangle
- Rhombus
- Trapezium
- Right angled trapezium
- Kite
Which of them are possible?
Hint 8
Five of the seven types of quadrilateral are possible. It is not possible to make a right angled trapezium or a kite.
Square
2x1 rectangle
√2 x 1 rectangle
Rhombus
TrapeziumHint 9
There are 8 types of shape with more than 4 sides that you might be asked to make:
- Pentagon
- Regular pentagon
- Hexagon
- Regular hexagon
- Heptagon
- Regular heptagon
- Octagon
- Regular octagon
Which of them are possible?
Hint 10
Three of the eight shapes with more than 4 sides are possible, the pentagon, the hexagon and the regular hexagon. It is not possible to make a regular pentagon and it is not possible to make a shape with more than 6 sides.
Pentagon
Hexagon
Regular hexagonPhase 3: Proving
As the result of the last section you might come up with these conjectures:
Conjecture 1: You can not make a shape with more than 6 sides.
Conjecture 2: You can not make a triangle that contains an angle greater than or equal to 90⁰.
Conjecture 3: No side length of a polygon is greater than √2.
Conjecture 4: Subject to Conjecture 2, the sides of any triangle may be up to √2 in length.
Conjecture 5: Squares and rectangles of any side length subject to Conjecture 3 can be made and an a by b rectangle can be made for any non-zero ratio a/b.
Conjecture 6: Trapezia can be made but none of them have right angles.
Conjecture 7: For n = 5 and 6, n-gons and regular n-gons exist.
Hint 11
What is it about the cube that limits the number of sides we can get on the new faces exposed by a plane cut?
Hint 12
Why is there any limit on the size of angles in a triangle made by plane cuts?
Hint 13
Why can't you get a right angled triangle?
Hint 14
How can you get an angle bigger than 90⁰ on a face produced by a plane cut of a cube?
Hint 15
Use the above hints to look at quadrilaterals.
Hint 16
Can you make a regular pentagon? Can you make many different ones?
Hint 17
Can you make one regular hexagon? Can you make many different ones?
Phase 4: Extending
The question now is can we take the problem any further. What Toni’s Tiara-like problems can you build around this one? It’s very important to think about such things. Extending problems is a fundamental way of working for mathematicians and students should know that (and it happens to be fun too). Let us suggest some related problems.
Some of these questions are more interesting than others. Many of them use similar techniques to the ones that we have used above.
- What shaped faces can be obtained from cutting a sphere?
- What shaped faces can be obtained from cutting a cuboid?
- What shaped faces can be obtained from cutting a tetrahedron?
- What shaped faces can be obtained from cutting an octahedron?
- What shaped faces can be obtained from cutting a decahedron?
- What shaped faces can be obtained from cutting a dodecahedron?
- Can we get all possible n-gons for n < 7 by cutting a cuboid?
- What solid do we need to cut in order to get all polygons for n < r?
