Литагент HarperCollins - The Ultimate Mathematical Challenge - Over 365 puzzles to test your wits and excite your mind

What are the numbers in this sum?

[SOLUTION]

53. A list of primes

Alice writes down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4 and 5 only once and using no other digits.

Which prime number must be in her list?

[SOLUTION]

54. Continue the pattern

The diagram shows the first three patterns in a sequence in which each pattern has a square hole in the middle.

How many small shaded squares are needed to build the tenth pattern in the sequence?

[SOLUTION]

55. How many codes?

Peter has a lock with a three-digit code. He knows that all the digits of his code are different, and that if he divides the second digit by the third and then squares his answer he will get the first digit.

What is the difference between the largest and smallest possible codes?

[SOLUTION]

56. A word product

What is the value of P + Q + R in the multiplication shown?

[SOLUTION]

Crossnumber 2

ACROSS

1. The highest common factor of 5 DOWN and 8 DOWN (2)

3. A prime factor of 2007 (3)

5. 3 DOWN plus the square root of 4 DOWN (3)

6. The product of three consecutive integers, two of which are prime (3)

7. One less than a multiple of 2 DOWN (3)

9. Five less than 14 ACROSS (4)

11. Seven more than the product of the digits of 22 ACROSS (2)

13. Three more than a triangular number (2)

14. 9 ACROSS plus five (4)

16. A square whose digit sum is three more than its square root (3)

18. Three times the product of two consecutive prime numbers (3)

21. The mean of 11 ACROSS and 21 ACROSS is 16 ACROSS (3)

22. Twice a prime number (3)

23. Two less than a square (2)

DOWN

1. Eight less than a multiple of nine (3)

2. A prime factor of 12 DOWN (2)

3. A Fibonacci number that is also a prime (3)

4. A square (2)

5. One less than twice a triangular number (3)

7. 10 DOWN minus three (4)

8. One third the product of three consecutive numbers, two of which are prime (3)

10. 7 DOWN plus three (4)

12. A number whose digit sum is equal to one of its factors (3)

15. The product of two consecutive prime numbers (3)

17. p 4+ 1, where p is prime (3)

19. The sum of 16 ACROSS and 3 ACROSS (3)

20. Six less than twice 13 ACROSS (2)

21. Fifteen plus the mean of 1 ACROSS and 11 ACROSS (2)

[SOLUTION]

Week 9

57. Three Tuesdays

Three Tuesdays of a month fall on even-numbered dates.

Which day of the week was the twenty-first day of the month?

[SOLUTION]

58. Crack the code

In a seven-digit numerical code, each group of four adjacent digits adds to 16 and each group of five adjacent digits adds to 19.

What is the code?

[SOLUTION]

59. Mr Bean’s fruit

Despite his name, Mr Bean likes to eat lots of fruit. He finds that four apples and two oranges cost £1.54 and that two oranges and four bananas cost £1.70.

How much would he have to pay if he bought one apple, one orange and one banana?

[SOLUTION]

60. Ali’s bookshelves

Ali is arranging the books on his bookshelves. He puts half his books on the bottom shelf and two-thirds of what remains on the second shelf. Finally, he splits the rest of his books over the other two shelves so that the third shelf contains four more books than the top shelf. There are three books on the top shelf.

How many books are on the bottom shelf?

[SOLUTION]

61. An unfair dice

I have an unfair dice that has probability of landing on a six, with all the other numbers equally likely. If the dice is thrown twice, what is the probability of obtaining a total score of ten?

[SOLUTION]

62. A room in Ginkrail

The town of Ginkrail is inhabited entirely by knights and liars. Every sentence spoken by a knight is true, and every sentence spoken by a liar is false. One day some inhabitants of Ginkrail were alone in a room and three of them spoke.

The first one said: ‘There are no more than three of us in the room. All of us are liars.’

The second said: ‘There are no more than four of us in the room. Not all of us are liars.’

The third said: ‘There are five of us in the room. Three of us are liars.’

How many people were in the room and how many liars were among them?

[SOLUTION]

63. Curious integers

In the following puzzle, each different capital letter represents a different digit. Thus ‘SEVEN’ represents a five-digit decimal number.

‘SEVEN’ is prime and, as one would expect, ‘SEVEN’ minus ‘THREE’ equals ‘FOUR’ .

Curiously, ‘FOUR’ is prime (as is ‘RUOF’ ) but ‘THREE’ is not prime. Another oddity is that ‘TEN’ is a square.

Find the values of ‘FOUR’ and ‘TEN’ .

[SOLUTION]

Week 10

64. Eight factors

A certain number has exactly eight factors including 1 and itself. Two of its factors are 21 and 35.

What is the number?

[SOLUTION]

65. A nonagon problem

The diagram shows a regular nine-sided polygon (a nonagon or an enneagon ) with two of the sides extended to meet at the point X .

What is the size of the acute angle at X ?

[SOLUTION]

66. How many primes?

Peter wrote a list of all the numbers that could be produced by changing one digit of the number 200.

How many of the numbers on Peter’s list are prime?

[SOLUTION]

67. Fill in the blanks

Sam wants to complete the diagram so that each of the nine circles contains one of the digits from 1 to 9 inclusive and each contains a different digit.

Also, the digits in each of the three lines of four circles must have the same total. What is this total?

[SOLUTION]

68. The school netball league

In our school netball league, a team gains a certain whole number of points if it wins a game, a lower whole number of points if it draws a game and no points if it loses a game.

After 10 games my team has won 7 games, drawn 3 and gained 44 points. My sister’s team has won 5 games, drawn 2 and lost 3.

How many points has her team gained?

[SOLUTION]

69. How many zogs?

The currency used on the planet Zog consists of bank notes of a fixed size differing only in colour. Three green notes and eight blue notes are worth 46 zogs; eight green notes and three blue notes are worth 31 zogs.