More questions to ponder… ***

1) How may millimetres in a kilometre? 

2) How many numbers between 1 and 100 do not have a 9 in them? 

3) Ten posts are placed 20 metres apart on a field.  What is the distance between the first and last post?

Deathly Probability Solution

The order of likelihood is:
1. Playing football
2. Snake bite
3. Falling down a manhole
4. Computer game exhaustion
5. Hippo attack
6. Struck by lightning
7. Hit by a falling coconut
8. Shark attack
9. Walking into a lamp post
10. Hit by a meteorite

Deathly Probability ***

Sometimes we can be influenced by things that aren't really facts.  For example, books and blockbuster films lead us to believe that sharks are very dangerous to humans.  In reality, however, more people are killed by hippos than sharks.  Try putting these causes of death in order of probability (from most likely to least likely).

• computer game exhaustion
• snake bite
• hippo attack
• walking into a lamp post
• falling down a manhole
• playing football
• hot by a falling coconut
• struck by lightning
• hit by a meteorite
• shark attack

Painted Cubes Solution

The 3 x 3 x 3 cube has:

- 1 cube right in the middle with zero sides painted

- 6 cubes with one side painted (1 in the centre of each of the six faces)

- 12 cubes with two sides painted (1 in the centre of each edge of the large cube)

-8 cubes with three sides painted (1 in each of the eight corners of the larger cube)

In general, for an n x n x n cube, think about how to count all the cubes with no sides painted.  Imagine removing the entire outer layer of small cubes.  You’ll be left with a cube in the centre, but each of its dimensions will be shrunk by 2 because a layer of cubes has been removed from both sides.  Now it is an n-2 by n-2 by n-2 cube so it has (n-2)³ little cubes with no sides painted.  For the cubes with one side painted, these are on the interior of each face.  By similar reasoning as above, this is an n-2 by n-s square, so there are (n-2)² of these for each of the six faces, so 6(n-2)² have one side painted.  For the cubes with two sides painted, these are along each of the twelve edges.  There are n-2 of these along each edge so there are a total of 12(n-2) cubes with two sides painted.  Finally, no matter how large the cube is, the only cubes with 3 sides painted are the corner cubes therefore there are always 8 cubes with three sides painted.

Profit Margin Solution

Dodgem use: 60% of 12 is 7.2
Number of sessions: 4 x 8 = 32
Fares per session: 32 x £2 = £64
£64 x 7.2 = £460.80
Earnings - Cost to run: £460.80 - £144 = £316.80
Profit = £316.80

Shaking Hands Solution

3 people = 3 handshakes
4 people = 6 handshakes
5 people = 10 handshakes

The answers are all triangular numbers.  If you do not know what triangular numbers are, go and look them up.

Painted Cubes ***

A 3 x 3 x 3 cube is painted red on the outside.  If it is broken up into 1 x 1 x1 – unit cubes, how many of these small cubes have 3 faces painted?  Two faces painted?  One face painted?  No faces painted? 

Hint: Imagine or draw a cube with each face looking like the 3 x 3 grid below.  (Unfortunately the picture I had of a cube will not show up here.)  You could even build one with small blocks!

  

Extend your thinking *****

What about if you start out with a larger cube?  Can you come up with a general rule for an n x n x n cube?

I would very much like to see any answers you are able to come up with!

Happy Easter,

Mrs Hopkins