Maths Challenge 1

Cymraeg

Maths Challenge 1


1: Rugby ExperimentPercentages and Statistics

As an experiment, Fred asked rugby fans at the end of two different matches for volunteers to attempt a drop-kick goal. He wanted to see if being on a home field was advantageous. The results are shown below

Table of results

Fred is happy to see that he was right, fans on their home field were better than the away fans. Why?

You will need to work out how many fans were successful for both home and away.

By looking at how many of each fan group were successful, he saw that home fans success was 45⁄150 (30%), whilst away fans success was only 34⁄150 (22.67%).

Logic

This is a riddle believed to have been created by Albert Einstein when he was just a child. This challenge can be overcome purely by logic.

There are five neighbouring houses, painted five different colours. A person with a different nationality lives in each house. The five house owners each drink a certain type of beverage, play a certain sport, and keep a certain pet. No owners have the same pet, play the same sport, or drink the same beverage.

Who owns the fish?

You will need to use the following facts to work this out.

  1. The Briton lives in the red house.
  2. The Swede keeps dogs as pets.
  3. The Dane drinks tea.
  4. The green house is on the left of the white house.
  5. The owner of the green house drinks coffee.
  6. The person who plays football rears birds.
  7. The owner of the yellow house plays baseball.
  8. The man living in the centre house drinks milk.
  9. The Norwegian lives in the first house.
  10. The man who plays volleyball lives next to the one who keeps cats.
  11. The man who keeps the horse lives next to the man who plays baseball.
  12. The owner who plays tennis drinks beer.
  13. The German plays hockey.
  14. The Norwegian lives next to the blue house.
  15. The man who plays volleyball has a neighbour who drinks water.

You will need to use this Logic Grid. If you do not know how to use a logic grid, you can visit this logic puzzle website for a tutorial

House 1 House 2 House 3 House 4 House 5
Nationality Norwegian Danish British German Swedish
Colour yellow blue red green white
beverage water tea milk coffee beer
Sport baseball volleyball football hockey tennis
Pet cat horse bird fish dog


So, the German owns the Fish

Sequences

Sequence 1

1, 4, 7, 10, 13 ...

What are the next two numbers in this sequence?

What number comes after 40?

Sequence 2

1, 3, 9, 27, 81 ...

What are the next two numbers in this sequence?

What number comes after 6,561?

Sequence 3

1, 3, 6, 10, 15 ...

What are the next two numbers in this sequence?

What number comes after 45?

Sequence 4

1, 8, 27, 64, 125 ...

What are the next two numbers in this sequence?

What number comes after 729?

Sequence 5

0, 1, 1, 2, 3, 5, 8 ...

What are the next two numbers in this sequence?

What number comes after 55?

Sequence 6

4, 9, 25, 49 ...

What are the next two numbers in the sequence?

What number comes after 961?

Sequence 1: What is the common difference between each number?
Sequence 2: What do you need to do to the previous number to get the next?
Sequence 3: Think triangles...
Sequence 4: Think cubes...
Sequence 5: This sequence involves adding 2 numbers together
Sequence 6: This involves squares of certain numbers

Sequence 1:
Add 3 each time:
1, 4, 7, 10, 13, 16, 19 ... 40, 43
Sequence 2:
Multiply by 3 each time:
1, 3, 9, 27, 81, 243, 729 ... 6561, 19683
Sequence 3:
Triangular numbers or +1, +2, +3, +4 .... etc:
1, 3, 6, 10, 15, 21, 28 ... 45, 55
Sequence 4:
Cubed numbers:
1, 8, 27, 64, 125, 216, 343 ... 729, 1000
Sequence 5:
Fibonacci Sequence (add the last number numbers together for the next):
0, 1, 1, 2, 3, 5, 8, 13, 21 ... 55, 89
Sequence 6:
Squares of prime numbers:
4, 9, 25, 49, 121, 169 ... 961, 1369

4: The Price of PizzaBasic Operations

Three house-mates ordered pizza. When it arrived the delivery man billed them £25. The house-mates each paid £10. The delivery man gave them five £1 coins in change. As they couldn't divide it equally, they took £1 each and tipped the delivery man £2. They then realised that they had each paid a total of £9 which meant they paid £27 for the pizza and then tipped the delivery man £2 bringing the total to £29. Where did the other £1 go?

How many times are they including the tip in their calculation

They paid £27 for the pizza and the tip. Then they received £3 change making a total of £30. Their reasoning is flawed because they've added the tip twice but not the change value.

5: The Crocodile ChallengeCalculus and Trigonometry (AS-Level)

This maths problem is from a Scottish Highers Exam by SQA (2015, Paper 2, Q8).

The Exam Question:

A crocodile is stalking prey located 20 metres further upstream on the opposite bank of a river.

Crocodiles travel at different speeds on land and in water.

The time taken for the crocodile to reach its prey can be minimised if it swims to a particular point, P, x metres upstream on the other side of the river as shown in the diagram.

There is a river, on the near bank, to the left, is a crocodile. On the far bank and 20 metres to the right of the crocodile, is a zebra.

The time taken, T, measured in tenths of a second, is given by

\[T(x) = 5\sqrt{36 + x^2} + 4(20 - x)\]

  1. Calculate the time taken if the crocodile does not travel on land.
  2. Calculate the time taken if the crocodile swims the shortest distance possible.
  3. Between these two extremes there is one value of x which minimises the time taken. Find the value of x and hence calculate the minimum possible time.

The equation gives you the width of the river, just look for the Pythagoras equation.

1. 10.4 seconds
2. 11 seconds
3. x = 8, so T = 9.8 seconds