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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 the red door likes rugby and is next to the Maths teacher.
The sum of all the classroom door numbers is 51.
The favourite sport of the teacher in the classroom named ‘Sphere’ is cricket.
The classroom called ‘Triangle’ is next door to the classrooms of the Art and Science teachers.
One of the classrooms has a white door.
155. Angles around a triangle
What is the value of a + b + c + d + e + f?
156. Einstein sees two clocks
Albert Einstein was standing on the station platform thinking about relativity when he noticed he could see two station clocks. Each clock was digital, showing only hours and minutes. He observed that the display on one clock changed to the next minute 10 seconds before the correct time, whereas the display on the other clock changed to the next minute 10 seconds after the correct time.
For what fraction of the time did both clocks show the same time?
157. Halving an annulus
The shaded region in the diagram, bounded by two concentric circles, is called an annulus.
The circles have radii 2 cm and 14 cm. The dashed circle divides the area of this annulus into two equal areas.
What is its radius?
158. How many pairs?
How many pairs of numbers (a, b) exist such that the sum a + b, the product ab and the quotient
