Sergeant's Orders ***

A sergeant stands too long in the sun and gets confused.  His troops are lined up facing north.  The he gives the order to “Turn right” 70 times, and his troops do so.  In which direction are the troops facing at the end?

More questions to ponder... Solutions

1. 1 km = 1000m = 1, 000, 000 mm

2. 180m (With 10 posts there are only 9 gaps between posts.)

3. 79  - Between 1 and 100 there are 98 numbers and 19 of those numbers have a 9 in them so there are 79 numbers which do not have a nine.

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