Читать книгу The Ultimate Mathematical Challenge: Over 365 puzzles to test your wits and excite your mind - Литагент HarperCollins USD, F. M. L. Thompson - Страница 21

Week 10

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?

[SOLUTION]

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?

[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?

[SOLUTION]

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?

[SOLUTION]

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?

[SOLUTION]

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?

[SOLUTION]

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?

[SOLUTION]