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

Ten different numbers (not necessarily integers) are written down. Any number that is equal to the product of the other nine numbers is then underlined.

At most, how many numbers can be underlined?

[SOLUTION]

105. Placing draughts

Barbara wants to place draughts on a board in such a way that the number of draughts in each row is equal to the number shown at the end of the row, and the number of draughts in each column is equal to the number shown at the bottom of the column. No more than one draught is to be placed in any cell.

In how many ways can this be done?

[SOLUTION]

Week 16

106. Square roots

How many of the numbers

are greater than 10?

[SOLUTION]

107. How many lines?

The picture shows seven points and the connections between them.

What is the least number of connecting lines that could be added to the picture so that each of the seven points has the same number of connections with other points?

(Connecting lines are allowed to cross each other.)

[SOLUTION]

108. Where in the list?

There are 120 different ways of arranging the letters U, K, M, I and C. All of these arrangements are listed in dictionary order, starting with CIKMU.

Which position in the list does UKIMC occupy?

[SOLUTION]

109. Eva’s sport

Two sportsmen (Ben and Filip) and two sportswomen (Eva and Andrea) − a speed skater, a skier, a hockey player and a snowboarder − had dinner at a square table, with one person on each edge of the square.

The skier sat at Andrea’s left hand.

The speed skater sat opposite Ben.

Eva and Filip sat next to each other.

A woman sat at the hockey player’s left hand.

Which sport did Eva do?

[SOLUTION]

110. Pedro’s numbers

Pedro writes down a list of six different positive integers, the largest of which is N . There is exactly one pair of these numbers for which the smaller number does not divide the larger.

What is the smallest possible value of N ?

[SOLUTION]

111. The speed of the train

A train travelling at constant speed takes five seconds to pass completely through a tunnel that is 85 m long, and eight seconds to pass completely through a second tunnel that is 160 m long.

What is the speed of the train?

[SOLUTION]

112. What is ‘pqrst’?

The digits p , q , r , s and t are all different.

What is the smallest five-digit integer ‘ pqrst ’ that is divisible by 1, 2, 3, 4 and 5?

[SOLUTION]

Crossnumber 4

ACROSS

2. A power of two (4)

5. A prime factor of 12345 (3)

6. Six more than a multiple of 13 ACROSS (3)

8. A cube (2)

10. The product of the digits of 25 ACROSS and also less than half of 23 ACROSS (2)

11. The mean of 4 DOWN, 8 ACROSS, 10 ACROSS, 13 ACROSS and 20 ACROSS and more than 3 DOWN (2)

13. A Fibonacci number (2)

14. A multiple of seven (3)

17. Eight less than a square (3)

19. Seven less than 26 DOWN (2)

20. A number that is greater than 3 DOWN and less than 27 DOWN (2)

22. An even number that is the sum of a square and a triangular number in two different ways (2)

23. A prime whose digits add up to five (2)

25. A square and a multiple of five (3)

28. A multiple of 14 that includes a two and an eight among its digits (3)

29. Nine more than a power of 20 ACROSS (4)

DOWN

1. One hundred and ninety-five less than a square (4)

2. One less than a Fibonacci number (3)

3. The highest common factor of 9 DOWN and 15 DOWN (2)

4. The sum of two powers of two (2)

6. (25 ACROSS) per cent of 24 DOWN (3)

7. The shortest side of a right-angled triangle whose longer sides are 24 DOWN and 25 ACROSS (3)

9. The square of a triangular number; also one less than a multiple of five (3)

12. A factor of 732, each of whose digits is a power of two (3)

15. Five multiplied by 3 DOWN (3)

16. An even square; also a multiple of 8 ACROSS (3)

17. A multiple of 17, the product of whose digits is a square multiplied by seven (3)

18. A multiple of nine (3)

21. A power of 21 (4)

24. A factor of 360 (3)

26. Seven more than 19 ACROSS (2)

27. A cube (2)

[SOLUTION]

Week 17

113. How many routes?

How many different routes are there from S to T that do not go through either of the points U and V more than once?

[SOLUTION]

114. What Rachel drinks

A bottle contains 750 ml of mineral water. Rachel drinks 50% more than Ross, and these two friends finish the bottle between them.

How much does Rachel drink?

[SOLUTION]

115. A magic product square

Place the numbers

in the squares of the grid, with one number in each square, so that the products of the numbers in the three rows, the three columns and the two diagonals are all equal to 1.

[SOLUTION]

116. What is the angle?

The diagram shows a regular hexagon PQRSTU , a square PUWX and an equilateral triangle UVW .

What is the size of angle TVU ?

[SOLUTION]

117. A sum of numbers

Consider the list of all four-digit numbers that can be formed using only the digits 1, 2, 3 and 4, with no repetitions.

What is the sum of all the numbers in this list?

[SOLUTION]

118. How many knights?

A group of 25 people consists of knights, serfs and damsels.

Each knight always tells the truth, each serf always lies, and each damsel alternates between telling the truth and lying.

When each of them was asked: ‘Are you a knight?’, 17 of them said ‘Yes’. When each of them was then asked: ‘Are you a damsel?’, 12 of them said ‘Yes’. When each of them was then asked: ‘Are you a serf?’, 8 of them said ‘Yes’.

How many knights are in the group?

[SOLUTION]

119. Crossing the river

Two adults and two children wish to cross a river. They make a raft, but it will carry only the weight of one adult or two children.

What is the minimum number of times the raft must cross the river to get all four people to the other side?

(Note: The raft may not cross the river without at least one person on board.)

[SOLUTION]

Week 18

120. Gluing cubes

A cube is made by gluing together a number of unit cubes face-to-face. The number of unit cubes that are glued to exactly four other unit cubes is 96.