## Hints

Puzzles problem solving

1 If you can figure it out, there is a simple mathematical formula for providing the answer. Start by working out how many different ways the man could travel from West to East if there was just one circular path.

3 Divide the pentagram into the following sections and analyse them.

5 The full name of one of the guests is Jane Morgan.

6 As in Example 3 at the beginning of this section, write down all the possible combinations remaining after a white ball has been drawn out, but remember there are two white balls to consider.

7 Write down the order of floors being visited by the lift. What sequence or sequences emerge?

8 Think of the 3x3 array of 9 squares as a magic number square in which each horizontal, vertical and corner-to-corner line add up to the same total.

9 Look at the odd numbers and the even numbers in the sequence separately. Write them down separately. Can you see a sequence occurring?

10 Study each word carefully. Does each reveal part of the message?

11 Look for a relationship between rows of squares.

12 The sum of all the numbers from 1-9 inclusive is 45. The two numbers in the outside circles must, therefore, be 3 and 4.

13 The chance of picking out just one apple which contains a worm is 4 in 50. This would leave just 49 apples in the bag and 3 containing a worm.

14 By joining the dots up in different ways, can a familiar sequence be produced?

15 Looking across and down, what progression is occurring in each row and column?

16 Number the separate segments from 1 to 13.

17 Work out a route back from the black ball to the white ball.

18 I hobbled half as far as I ran.

19 Concentrate on numbers in lines across.

### 20 Look at the centre of each word.

21 Analyse how many different letters appear in the list of words. If, for example, nine different letters appear, you are looking to form a nine-letter word with these letters, i.e. a nine-letter word that does not repeat a letter.

22 If you are looking for one group of letters that is the odd one out, you must first find what the remaining groups of letters have in common.

23 It is a 1 in 26 chance that the first card will be the letter P.

24 The phrase you are looking for contains 4 x 4-letter words.

25 Look at alternate squares in each row and column.

Numerical problem solving

3 To solve this problem it is necessary to work out how many students studied one language only.

One way of doing this efficiently and quickly is to set up a Venn diagram.

Now see if you can determine the answer from the information you have been given.

4 Work out a hypothetical round of golf with 6 wins to Geoff, 4 wins to Harry and 2 holes tied.

The order of winning makes no difference to the outcome.

5 First, consider the numbers between 200 and 300 that have different factors. For example, 240 fingers could mean:

20 aliens with 12 fingers or 12 aliens with 20 fingers or 10 aliens with 24 fingers or 24 aliens with 10 fingers

One way of doing this efficiently and quickly is to set up a Venn diagram.

However, this is not a unique answer so it must be eliminated, as must all numbers having different factors.

Carry out a similar analysis with prime numbers. What conclusion do you arrive at?

6 If 3 lied, 3 told the truth.

Analyse my friends' answers with / ticks Only 3 / will give the correct month.

7 You must calculate the amount of money taken on each horse which will give a margin of 15% on the money

Calculate the amount to be laid on each horse to give a return of £100.

If the total is £115, that means a mark-up of £15 if the bookmaker balances his books.

8 The centre of the coin must fall within a square within the square.

Now try to calculate the odds using this information.

staked.

sttl

9 It would appear that the last number 7, should be an 8 if the puzzle is:

72 + 27 = 99 27 + 18 = 45 18 + 21 = 39 21 + 15 = 36 15 + 13 = 28

13 + 7 = (20?) (21?) so the numbers are produced in a different way.

10 To solve this problem you must find the lowest common multiple (LCM), that is, the smallest number that the number of cog teeth can be divided into.

The answer is this number divided by the teeth on the largest cog.

12 Use Pythagorus' theorem.

13 Try to find a formula which will provide the answer to this and similar problems.

Calculate the yards travelled for 100-yd maze, 10 squares. Then calculate the yards travelled for an 80-yd maze. This then will give the formula for even squares.

Then calculate a 90-yd square, this will give further parts to the formula.

14 For the puzzle to work, 1,111,111 must have only two factors apart from itself and unity.

Find the factors, they are both prime numbers.

Each factor ends in 9, one factor is under 250.

15 Analyse the numbers 8 to 100, split them up each time into the truth and lies, and find the only unique number that answers the questions.

16 Factorise 6591, and find 7 prime numbers which when multiplied together = 6591.

18 The left-hand side of the equation is divisible by 9 (32).

A number is divisible by 9 exactly, only when the sum of its digits is also exactly divisible by 9.

19 The odds of drawing the first correct number is one chance in 49.

The odds of drawing the second correct number is one chance in 48.

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