123. How many pairs?
How many pairs of digits (p, q) are there so that the five-digit integer ‘p869q’ is a multiple of 15?
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?
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:
| Yellows | 10 |
| Reds | 9 |
| Greens | 4 |
| Blues | 3 |
| Pinks | 1 |
How many of the matches resulted in a draw?
What were the results of the Greens’ matches against the other teams?
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?
See Shuttle Challenge 1 for 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 cm2.
What is the area, in cm2, 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.
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?
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?
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?
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?
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?
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