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

How many tests were in the series?

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140. Three primes

Find all positive integers p such that p , p + 8 and p + 16 are all prime.

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Crossnumber 5

ACROSS

1. The cube of a square (5)

4. Eight less than 5 DOWN (3)

6. One less than a multiple of seven (3)

7. A prime factor of 20 902 (4)

10. A number whose digits successively decrease by one (3)

12. Sixty per cent of 20 DOWN (3)

14. A multiple of seven (3)

15. A multiple of three whose digits have an even sum (3)

16. The square of a square (3)

17. A prime that is one less than a multiple of six (3)

19. Eleven more than a cube (4)

22. A number all of whose digits are the same (3)

24. A number that leaves a remainder of eleven when divided by thirteen (3)

25. The square of a prime; the sum of the digits of this square is ten (5)

DOWN

1. A number with an odd number of factors (3)

2. Four less than a triangular number (3)

3. The square root of 9 DOWN (2)

4. A factor of 12 ACROSS (3)

5. The longest side of a right-angled triangle whose shorter sides are 3 DOWN and 4 ACROSS (3)

8. A Fibonacci number (5)

9. The square of 3 DOWN (4)

11. Three more than an even cube (5)

13. A prime factor of 34567 (4)

17. The mean of 10 ACROSS, 16 ACROSS, 18 DOWN, 20 DOWN and 21 DOWN (3)

18. A power of eighteen (3)

20. Two less than 22 ACROSS (3)

21. A number whose digits are those of 12 ACROSS reversed (3)

23. A multiple of twenty-three (2)

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Week 21

141. Joey’s and Zoë’s sums

Joey calculated the sum of the largest and smallest two-digit numbers that are multiples of three. Zoë calculated the sum of the largest and smallest two-digit numbers that are not multiples of three.

What is the difference between their answers?

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142. When is the party?

Six friends are having dinner together in their local restaurant. The first eats there every day, the second eats there every other day, the third eats there every third day, the fourth eats there every fourth day, the fifth every fifth day and the sixth eats there every sixth day. They agree to have a party the next time they all eat together there. In how many days’ time is the party?

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143. A multiple of 11

The eight-digit number ‘1234 d 678’ is a multiple of 11.

Which digit is d ?

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144. Two squares

ABCD is a square. P and Q are squares drawn in the triangles ADC and ABC , as shown.

What is the ratio of the area of the square P to the area of the square Q ?

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145. Proper divisors

Excluding 1 and 24 itself, the positive whole numbers that divide into 24 are 2, 3, 4, 6, 8 and 12. These six numbers are called the proper divisors of 24.

Suppose that you wanted to list in increasing order all those positive integers greater than 1 that are equal to the product of their proper divisors. Which would be the first six numbers in your list?

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146. Kangaroo game

In the expression

the same letter stands for the same non-zero digit and different letters stand for different digits.

What is the smallest possible positive integer value of the expression?

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147. A game with sweets

There are 20 sweets on the table. Two players take turns to eat as many sweets as they choose, but they must eat at least one, and never more than half of what remains. The loser is the player who has no valid move.

Is it possible for one of the two players to force the other to lose? If so, how?

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Week 22

148. A 1000-digit number

What is the largest number of digits that can be erased from the 1000-digit number 201820182018 … 2018 so that the sum of the remaining digits is 2018?

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149. Gardeners at work

It takes four gardeners four hours to dig four circular flower beds, each of diameter four metres.

How long will it take six gardeners to dig six circular flower beds each of diameter six metres?

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150. Overlapping squares

The diagram shows four overlapping squares that have sides of lengths 5 cm, 7 cm, 9 cm and 11 cm.

What is the difference between the total area shaded grey and the total hatched area?

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151. What can T be?

Each of the numbers from 1 to 10 is to be placed in the circles so that the sum of each line of three numbers is equal to T . Four numbers have already been entered.

Find all the possible values of T .

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152. Increases of 75%

Find all the two-digit numbers and three-digit numbers that are increased by 75% when their digits are reversed.

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153. Three groups

For which values of the positive integer n is it possible to divide the first 3 n positive integers into three groups each of which has the same sum?

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154. A board game

Two players, X and Y , play a game on a board that consists of a narrow strip that is one square wide and n squares long. They take turns in placing counters that are one square wide and two squares long on unoccupied squares on the board.

The first player who cannot place a counter on the board loses. X always plays first, and both players always make the best available move.

Who wins the game in the cases where n = 2, 3, 4, 5, 6, 7 and 8?

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Logic Challenge 3

Five teachers work in a school. By using the clues in the statements below, you need to work out what subject the teacher teaches and what sport they like, and information about their classroom, including the room name and number, and the colour of the classroom door.

Fill in the answer grid with information about each teacher.

Mr Smith teaches Art.

The Science teacher is in the classroom called ‘Square’.

History is taught by Miss Jones.

The favourite sport of Mr Henry does not involve a ball.

Mrs Talbot’s classroom door is coloured yellow and is next to the classroom that has the largest single-digit prime number as its number.

The Maths teacher has the largest classroom door number.

The favourite sport of the teacher in the classroom called ‘Triangle’ is netball.

Miss Jones’s door is coloured orange.

The teacher in the classroom called ‘Circle’ is next to the Maths teacher.

English is taught in the classroom called ‘Sphere’.

The classroom door numbered 3 is next to the teacher who is next to the teacher whose favourite sport is jogging.

Football is the favourite sport of the teacher in the classroom called ‘Cylinder’.