You might be familiar with divisibility rules such as “A number is divisible by 3 if the sum of the digits is divisible by 3”. You can see some more here.
Our investigation today was – can we develop a set of divisibility rules for binary numbers?
Divisible by 2
This was straightforward. If the rightmost (unit) digit is 0 then the number is divisible by 2.
e.g. The number 6 is divisible by 2
but the number 7 is not
Can you extend this logic to create a divisibility rule for powers of 2 (e.g. 4, 8, 16 …)? Solution
This activity came from watching this Numberphile video . A worksheet of the activities below is available here.
Suppose that we want to colour the whole plane so that any two points at distance 1 from each other would have different colors.
You can imagine this as walking on a tiled floor in jumps of 1 and each jump has to be onto a different colour.
What is the minimum number of colours we would need to use?
Finding an upper bound to the number of colours needed
Task 1
It doesn’t have to be a regular tiling, but would 4 colours work if we coloured the plane like above? How big would the squares have to be?
Task 2
Find a number of colours that would work for a square tiling. How big would the squares have to be?
Task 3
Show that we would need at most 7 colours using a different shape tile (isometric paper will be helpful!).
Finding an lower bound to the number of colours needed
Task 4
Use this diagram to show that we would need at least 3 colours to colour the plane.
Task 5
Create a graph of points to show that we need at least 4 colours.
Further reading
This problem is called the Hadwiger–Nelson problem and was stated in 1950. It was recently (2018!) proved that you need at least 5 colours by an amaetur mathematician who was working on it as a break from his job as a maverick biologist intent on extending the human lifespan! The middle dot in the picture below has to be white rather than red, green, blue or yellow like the rest. It is still an open problem to find the smallest number of colors. It could now be 5, 6 or 7 and nobody knows.
Today we used inspiration from Nrich and mathsisfun.com to explore the Platonic solids on Polypad.
Task 1 – Dodecahedron
Go to https://polypad.amplify.com/p#polygons and make a net of a dodecahedron. You can keep adding pentagons together and then highlight them all and click Fold Net.
If it is not quite right, click Unfold Net and then add/remove pentagons and try again.
Task 2 – Angle deficit
The angle deficit at a vertex of a polyhedron is a measure of how far short each angle sum is from 360°. For example, in a dodecahedron, three pentagons with interior angles of 108° meet at each vertex, so the angle sum is 324° and the angle deficit is 36°:
The total angle deficit is the sum of the deficits at each vertex.
Can you fill in the table for the Dodecahedron:
Platonic Solids
Regular polygons
Faces
Vertex form
Angle Sum
Angle Deficit
Number of Vertices
Total Angle Deficit
Cube
Squares
Tetrahedron
Triangle
Octahedron
Triangle
Icosahedron
Triangle
Dodecahedron
Pentagon
5,5,5
324
36
Task 3 – Make the nets and fill in the table for the other four Platonic Solids. These are 3D shapes where the faces are congruent regular polygons and the same number of faces meet at each vertex.
Task 4 – Find all the Archimedean Solids which are formed by two or more types of regular polygons, each with the same side length and such that each vertex has the same pattern of polygons around it.
20 is a practical number because it could be used to design a set of weights where all of its factors are the weights and you can weigh all the numbers from 1 to 20 using each weight only once.
The factors of 20 are: 1, 2, 4, 5, 10 and 20
And you can make the numbers 1 to 20 using combinations of these numbers (each number can be used a maximum of once):
So 20 is a practical number
Of course all powers of 2 are practical numbers, but can you find some others?
Do they have anything in common?
Are there an infinite number of practical numbers?
Once you’ve found some can you use them to generate more?
Using these facts:
The product of two practical numbers is also a practical number.
More strongly, the least common multiple of any two practical numbers is also a practical number.
if n is a practical number and d is one of its divisors then d must also be a practical number.
So for example with the primitive practical numbers 1 and 2 you can generate 2*2=4
So 4 is not a primitive number as it can be generated from previous ones.
Can you circle all the practical numbers in this grid, using a different colour for primitive practical numbers?
Solution can be found here. To read more about Practical Numbers check out Wikipedia and this talk.
