63. Curious integers
In the following puzzle, each different capital letter represents a different digit. Thus ‘SEVEN’ represents a five-digit decimal number.
‘SEVEN’ is prime and, as one would expect, ‘SEVEN’ minus ‘THREE’ equals ‘FOUR’.
Curiously, ‘FOUR’ is prime (as is ‘RUOF’) but ‘THREE’ is not prime. Another oddity is that ‘TEN’ is a square.
Find the values of ‘FOUR’ and ‘TEN’.
64. Eight factors
A certain number has exactly eight factors including 1 and itself. Two of its factors are 21 and 35.
What is the number?
65. A nonagon problem
The diagram shows a regular nine-sided polygon (a nonagon or an enneagon) with two of the sides extended to meet at the point X.
What is the size of the acute angle at X?
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?
