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

How many unit cubes are glued to exactly five other unit cubes?

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121. Mr Gallop’s ponies

Mr Gallop has two stables that each initially housed three ponies. His prize pony, Rein Beau, is worth £250 000. Rein Beau usually spends his day in the small stable, but when he wandered across into the large stable, Mr Gallop was surprised to find that the average value of the ponies in each stable rose by £10 000.

What is the total value of all six ponies?

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122. Making a square

I have two types of square tile. One type has a side length of 1 cm and the other has a side length of 2 cm.

What is the smallest square that can be made with equal numbers of each type of tile?

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123. How many pairs?

How many pairs of digits ( p, q ) are there so that the five-digit integer ‘ p 869 q ’ is a multiple of 15?

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124. What is the product?

Lucy wants to put the numbers 2, 3, 4, 5, 6 and 10 into the circles so that the products of the three numbers along each edge are the same, and as large as possible.

In how many ways can this be done?

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125. A five-team league

Five teams played in a competition and every team played once against each of the other four teams. Each team received three points for a match it won, one point for a match it drew and no points for a match it lost.

At the end of the competition the points were as follows:

How many of the matches resulted in a draw?

What were the results of the Greens’ matches against the other teams?

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126. A mini crossnumber

The solution to each clue of this crossnumber is a two-digit number, not beginning with zero.

Across

1. A triangular number

3. A triangular number

Down

1. A square

2. A multiple of 5

In how many different ways can the crossnumber be completed correctly?

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Shuttle Challenge 2

See Shuttle Challenge 1for how the Shuttle works.

Question 1

What is the value of ?

Question 2

[ A is the answer to Question 1.]

The number A is an example of a palindromic integer – one that is unchanged when the order of its digits is reversed.

How many palindromic integers are there from 300 to A inclusive?

Question 3

[ A is the answer to Question 2.]

The diagram shows a triangle drawn on a square grid made up of nine smaller squares.

The area of the shaded triangle is A cm 2.

What is the area, in cm 2, of one of the smaller squares?

Question 4

[ A is the answer to Question 3.]

Write the number A as a word in the gap shown in the following sentence.

Out of the first __________ letters in this sentence, what fraction is vowels?

Now answer the question.

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Week 19

127. Coins in a frame

The diagram shows 10 identical coins that fit exactly inside a wooden frame. As a result, each coin is prevented from sliding.

What is the largest number of coins that may be removed so that each remaining coin is still unable to slide?

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128. Cheetah v. snail

In a sponsored ‘Animal Streak’, the cheetah ran at 90 kilometres per hour, while the snail slimed along at 20 hours per kilometre. The cheetah kept going for 18 seconds.

Roughly how long would the snail take to cover the same distance as the cheetah?

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129. Memorable phone numbers

A new taxi firm needs a memorable phone number. They want a number that has a maximum of two different digits. Their phone number must start with the digit 3 and be six digits long.

How many such numbers are possible?

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130. How many games?

Cleo played 40 games of chess and scored 25 points.

A win counts as one point, a draw counts as half a point, and a loss counts as zero points.

How many more games did she win than lose?

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131. How many draughts?

Barbara wants to place draughts on a 4 × 4 board in such a way that the number of draughts in each row and in each column are all different. (She may place more than one draught in a square, and a square may be empty.)

What is the smallest number of draughts that she would need?

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132. To and from Jena

In a certain region these are five towns: Freiburg, Göttingen, Hamburg, Ingolstadt and Jena.

One day, 40 trains each made a journey, leaving one of these towns and arriving at another.

10 trains travelled either to or from Freiburg.

10 trains travelled either to or from Göttingen.

10 trains travelled either to or from Hamburg.

10 trains travelled either to or from Ingolstadt.

How many trains travelled either to or from Jena?

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133. Largest possible remainder

What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits?

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Week 20

134. n th term n

The first term of a sequence of positive integers is 6. The other terms in the sequence follow these rules:

if a term is even then divide it by 2 to obtain the next term;

if a term is odd then multiply it by 5 and subtract 1 to obtain the next term.

For which values of n is the n th term equal to n ?

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135. How many numbers?

Rafael writes down a five-digit number whose digits are all distinct, and whose first digit is equal to the sum of the other four digits.

How many five-digit numbers with this property are there?

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136. A square area

A regular octagon is placed inside a square, as shown.

The shaded square connects the midpoints of four sides of the octagon.

What fraction of the outer square is shaded?

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137. ODD plus ODD is EVEN

Find all possible solutions to the ‘word sum’ shown.

Each letter stands for one of the digits 0−9 and has the same meaning each time it occurs. Different letters stand for different digits. No number starts with a zero.

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138. Only odd digits

How many three-digit multiples of 9 consist only of odd digits?

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139. How many tests?

Before the last of a series of tests, Sam calculated that a mark of 17 would enable her to average 80 over the series, but that a mark of 92 would raise her average mark over the series to 85.