The Ultimate Mathematical Challenge: Over 365 puzzles to test your wits and excite your mind. Литагент HarperCollins USD
[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?
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?
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?
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.
How many zogs are two green notes and three blue notes worth?
70. How many V-numbers?
A three-digit integer is called a ‘V-number’ if the digits go ‘high-low-high’ – that is, if the tens digit is smaller than both the hundreds digit and the units (or ‘ones’) digit.
How many three-digit ‘V-numbers’ are there?
In the Shuttle rounds of the Team and Senior Team Maths Challenges, each team of four students is divided into two pairs who sit at opposite ends of a table. One pair tackles questions 1 and 3; the other pair attempts questions 2 and 4. The numerical answer to question 1 is passed across the table to the other pair who need it to answer question 2, and so on. The answer that is passed on is called A in the subsequent question.
The teams have eight minutes to answer all four questions. They get bonus marks if they answer all the questions correctly within six minutes.
How long will it take you?
Question 1
What is the value of (42 + 52) × 72?
Question 2
[A is the answer to Question 1.]
At which number will the minute hand of a clock be pointing to (A + 1) minutes after midnight?
Question 3
[A is the answer to Question 2.]
John has three sticks that he has formed into a triangle. The length of each stick is a whole number of centimetres.
The length of one of the sticks is (A + 1) cm, and the length of another of the sticks is (A − 1) cm.
How many different possibilities are there for the length of John’s third stick?
Question 4
[A is the answer to Question 3.]
A pyramid with a polygonal base has A faces.
How many edges does the pyramid have?
71. A magic square
In a magic square, each row, each column and both main diagonals have the same total.
In the partially completed magic square shown, what number should replace N?
72. Fly, fly, fly away
In this addition sum, each letter represents a different non-zero digit.
What are the numbers in this sum?
73. What is the units digit?
Catherine’s computer correctly calculates
What is the units digit of its answer?
74. Minnie’s training
After a year’s training, Minnie Midriffe increases her average speed in the London Marathon by 25%.
By what percentage did her time decrease?
75. Telling the truth
The Queen of Hearts always lies for the whole day or always tells the truth for the whole day.
Which of these statements can she never say?
A. ‘Yesterday, I told the truth.’
B. ‘Yesterday, I lied.’
C. ‘Today, I tell the truth.’
D. ‘Today, I lie.’
E. ‘Tomorrow, I shall tell the truth.’
76. What is the unshaded area?
Eight congruent semicircles are drawn inside a square of side length 4.
Each semicircle begins at a vertex of the square and ends at a midpoint of an edge of the square.
What is the area of the unshaded part of the square?