The Ultimate Mathematical Challenge: Over 365 puzzles to test your wits and excite your mind. Литагент HarperCollins USD
ABCD is a square. P and Q are squares drawn in the triangles ADC and ABC, as shown.
What is the ratio of the area of the square P to the area of the square Q?
145. Proper divisors
Excluding 1 and 24 itself, the positive whole numbers that divide into 24 are 2, 3, 4, 6, 8 and 12. These six numbers are called the proper divisors of 24.
Suppose that you wanted to list in increasing order all those positive integers greater than 1 that are equal to the product of their proper divisors. Which would be the first six numbers in your list?
146. Kangaroo game
In the expression
the same letter stands for the same non-zero digit and different letters stand for different digits.
What is the smallest possible positive integer value of the expression?
147. A game with sweets
There are 20 sweets on the table. Two players take turns to eat as many sweets as they choose, but they must eat at least one, and never more than half of what remains. The loser is the player who has no valid move.
Is it possible for one of the two players to force the other to lose? If so, how?
148. A 1000-digit number
What is the largest number of digits that can be erased from the 1000-digit number 201820182018 … 2018 so that the sum of the remaining digits is 2018?
149. Gardeners at work
It takes four gardeners four hours to dig four circular flower beds, each of diameter four metres.
How long will it take six gardeners to dig six circular flower beds each of diameter six metres?
150. Overlapping squares
The diagram shows four overlapping squares that have sides of lengths 5 cm, 7 cm, 9 cm and 11 cm.
What is the difference between the total area shaded grey and the total hatched area?
151. What can T be?
Each of the numbers from 1 to 10 is to be placed in the circles so that the sum of each line of three numbers is equal to T. Four numbers have already been entered.
Find all the possible values of T.
152. Increases of 75%
Find all the two-digit numbers and three-digit numbers that are increased by 75% when their digits are reversed.
153. Three groups
For which values of the positive integer n is it possible to divide the first 3n positive integers into three groups each of which has the same sum?
154. A board game
Two players, X and Y, play a game on a board that consists of a narrow strip that is one square wide and n squares long. They take turns in placing counters that are one square wide and two squares long on unoccupied squares on the board.
The first player who cannot place a counter on the board loses. X always plays first, and both players always make the best available move.
Who wins the game in the cases where n = 2, 3, 4, 5, 6, 7 and 8?
Five teachers work in a school. By using the clues in the statements below, you need to work out what subject the teacher teaches and what sport they like, and information about their classroom, including the room name and number, and the colour of the classroom door.
Fill in the answer grid with information about each teacher.
Mr Smith teaches Art.
The Science teacher is in the classroom called ‘Square’.
History is taught by Miss Jones.
The favourite sport of Mr Henry does not involve a ball.
Mrs Talbot’s classroom door is coloured yellow and is next to the classroom that has the largest single-digit prime number as its number.
The Maths teacher has the largest classroom door number.
The favourite sport of the teacher in the classroom called ‘Triangle’ is netball.
Miss Jones’s door is coloured orange.
The teacher in the classroom called ‘Circle’ is next to the Maths teacher.
English is taught in the classroom called ‘Sphere’.
The classroom door numbered 3 is next to the teacher who is next to the teacher whose favourite sport is jogging.
Football is the favourite sport of the teacher in the classroom called ‘Cylinder’.
The teacher in the classroom called ‘Square’ is next to the classrooms with the green and orange doors.
The classroom called ‘Cylinder’ is next to the classroom of the Science teacher.
English is taught in the classroom with the ‘unlucky’ prime number.
The classroom door of the Maths teacher is green.
The classroom numbers of Mrs Talbot and Mrs Richard add up to 30.
All the classroom door numbers are prime numbers.
The English teacher is next to a classroom that is next to the classroom of the Science teacher.
The teacher in the classroom with