Toni's Tiara: Teachers' Notes
Introduction
Despite the fact that there is a tiara at stake, this problem is all about cutting a section through a cube. What possible shapes will the new faces on the cut cube have?
You can think of this as being the shape that you get on the exposed face when a cube is cut in two by a plane. Now, to get some idea of the possible answers to this, your students can go out and make a cube of playdough and then cut through it in a multitude of possible ways or they can go to the ‘Experiment’ button. This will provide them with a red cube. The white outline in the diagram is the shape they can get on the exposed face by cutting the cube with a plane (not explicitly shown in the diagram). If they push the cube itself they can make it rotate to expose other white shapes. If they move the arrow on the vertical scale on the left this is equivalent to moving the cube backwards and forwards through the plane of the cut.
After some experimenting they might get some idea of how the faces might look. They can then tackle the Toni’s Tiara problem. But the demons that control the program are tenacious and your students will have to justify all the choices they make with respect to the shapes on the chest that holds Toni’s tiara. So it’s a good idea to be sure what shapes can be made before tackling the retrieval of Toni’s tiara.
About the problem
Hopefully the problem is clear. Students need to find the different possible shapes that can be exposed by slicing through a cube. This will probably be discovered by a process of trial and error while playing with the red cube in the ‘Experiment’ section. Once a few polygons have been discovered the student should be encouraged to be more specific and try to see which triangles can be produced by cutting a cube and then which quadrilaterals, and so on. It’s even important to be able to determine the largest value of n for which an n-gon can be found.
Once the student has a ‘guessed’ list the things on the list become conjectures. From there it’s a matter of proof. Some proofs are straightforward but all should be understandable by a student in the middle years of secondary school. Some other proofs are hard and will not be accessible to most students no matter how good they are at their normal class work.
Some possible solutions
Very quickly it can be seen from the ‘Experiment’ mode that a number of polygons can be made. We can certainly get triangles, squares, trapezia, pentagons and hexagons. It looks as if we might be able to get equilateral triangles. So what about right angled, obtuse, acute and scalene triangles?
Can you also get equilateral triangles of every side length? What does that actually mean? Surely it all depends on the cube? So from here on let’s assume that the cube is 1 unit by 1 unit by 1 unit. Since the diagonal of a face is therefore √2 this will no doubt limit the size of the sides of any triangle we produce by lopping a corner off of the cube.
Which leads on to what size and shape of quadrilaterals we can find and so on.
And while we are at it, can you get a shape with more than 6 sides?
There are, in fact, so many questions that we won’t answer them all here. Indeed we won’t even ask them. But we will try to find answers to all of the questions that are needed to retrieve Toni’s loot. We provide this list here of shapes that you may be challenged to make:
Triangles
- right angled triangle
- equilateral triangle
- isosceles triangles
- scalene triangle
- obtuse triangle
Quadrilaterals
- square
- 2 x 1 rectangle
- √2 x 1 rectangles (Note that in the glossary in the game the 2 is missing from the √2)
- kite
- rhombus
- trapezium
- right trapezium
Shapes with more than 4 sides
- Regular pentagon
- Hexagon
- Regular hexagon
- Heptagon
- Regular heptagon
- Octagon
- Regular octagon
All this cerebral effort may well lead to the following 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.
Clearly some our conjectures go further than we need for the problem. Some proofs are available on the Toni's Tiara: Proof page.
Extension
The question now is can we take the problem any further. What Toni’s Tiara-like problems can we build around this one? It’s very important to have this discussion with students at all levels. Extending problems is a fundamental way of working for mathematicians and students should know that. However, the obvious extensions are not as exciting as the cube
- 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 a 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 polygons with 6 or fewer sides by cutting a cuboid?
- What solid do we need to cut in order to get all polygons for n < r?
