Fig.1 below shows three regular polygons meeting at a central point whilst Fig. 2 shows a cuboid whose side lengths were chosen to correspond to the number of sides in the polygons in Figure 1. In other words they are both based on the triplet (4,8,8).
It turns out that the volume and surface area of the cuboid in Fig. 2 are numerically equal. We call such objects 'Equable cuboids'.
Is this a coincidence or does every triplet of three regular polygons that fit round a point always generate equable cuboids?
Take every integer power (greater than the first power) of every positive integer greater than 1 and add the reciprocals together.
What do you get?
In the news on 28^{th} September 2008 it was announced that a new prime number had been discovered with 12,978,189 digits.
It is of the form 2^{p}–1 where is a p prime number.
How might you find the value of p?
How are the areas of the three circles linked?
What about other regular polygons?
It is possible for square wheel to move such that the axle remains at constant height, provided the surface it is "rolling" on is a specific shape.
Can you show that the shape of the surface is not an arc of a circle?
What is the shape of the surface?
On 30 cards I write 30 different numbers, shuffle them thoroughly and place them face down in a pile on a table. You have no idea how big any of my numbers are. You turn over one card at a time, look at the number and decide whether you want to stick with that card or discard it and turn over the next card. Once a card has been discarded you cannot go back to it. What strategy should you use in order to maximise your chances of selecting the card containing the biggest number?
A possible strategy: Do not keep any of the first 10 cards you turn over but remember the biggest number and call this number N. From the 11th card on, as soon as you see a number bigger than N, stop there. Of course, you might not see another number bigger than N in which case you will end up turning over all the cards without 'winning'.
With this strategy, what is the probability that you will select the biggest number?
Find a cubic with a single stationary point at x=a (i.e. dy/dx=0 has a repeated real root). Plot the roots of the cubic and x=a on an Argand diagram.
What do you notice?
What happens if the restriction of a single stationary point is removed?
Please see our MEI Conference 2008 Page.
x |
-2 |
-1 |
0 |
1 |
2 |
y |
-2 |
-1 |
0 |
1 |
2 |
The curve y = x^{5} – 5x^{3} + 5x passes through these points.
What are the x-coordinates of the stationary points on this curve?
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 6^{2}
Are there infinitely many triangle numbers that are also square numbers?
There are a number of mathematical podcasts/webcasts on the internet. The following are recommended:
What is the smallest positive integer that cannot be expressed using exactly three 2s and any mathematical operations you wish?
Here are some examples:
A hint:
Can you adapt the following for other numbers?