Weekly Problem 1 - 2004
Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?
Rotating a pencil twice about two different points gives surprising results...
Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?
Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radii.
You may have seen magic squares before, but can you work out the missing numbers on this magic star?
This magic square has only been partially completed. Can you still solve it...
Four wiggles equal three woggles. Two woggles equal five waggles. Six waggles equal one wuggle. Using these, can you work out which of four values is the smallest?
You have already used Magic Squares, now meet an Anti-Magic Square. Its properties are slightly different, but can you still solve it...
Multiply a sequence of n terms together. Can you work out when this product is equal to an integer?
Place equal, regular pentagons together to form a ring. How many pentagons will be needed?
Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?
The equation $x^2+2=y^3$ looks nearly quadratic. What integer solutions can you find?
Draw two intersecting rectangles on a sheet of paper. How many regions are enclosed? Can you find the largest number of regions possible?
Find a number between 100 and 999 that has its middle digit equal to the sum of the other two digits. Can you find all possibilities?
Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.
What is the smallest possible number of pairs of students who are awarded the same mark in an exam out of 100 if there are 120 students altogether?
ABCDEFGHI is a regular nine-sided polygon (called a 'nonagon' or 'enneagon'). What is the size of the angle FAE ?
An irregular hexagon can be made by cutting the corners off an equilateral triangle. How can an identical hexagon be made by cutting the corners off a different equilateral triangle?
Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees. What is the greatest number of sides the polygon could have?
The diagram shows a regular dodecagon. What is the size of the marked angle?
Poisoned ivy cost £2 each, deadly nightshade cost £9 each and triffids cost £12 each. Jasmine buys twenty plants in total and spends exactly £120. How many triffids did she buy?
The diagram shows a parallelogram inside a triangle. The marked lengths are equal. What is the value of x?
In the diagram, angle MON = 130 degrees. The reflection of OP in OM is OQ and the reflection of OP in ON is OR. What is the size of angle QOR?
The diagram shows an equilateral triangle inside a rhombus. The sides of the rhombus are equal in length to the sides of the triangle. What is the value of x?
I walk to the bike shop at 3 miles per hour, and I cycle home at 12 miles per hour. What is my average speed, in miles per hour, for the time that I am actually on the route?
Two thirds of five sixths of a number X is the same as three quarters of four fifths of a number Y. What is the value of X/Y as a fraction in its lowest terms?
A square piece of wood slides once around another square. Through what distance does a point on the square move?
Which of the following statements is false? (a) An octagon has 20 diagonals. (b) A hexagon has 9 diagonals. (c) A hexagon has 4 more diagonals than a pentagon...
If a decimal clock started from 0:00 at midnight, what time would it show when an ordinary clock showed 6 o'clock the following morning?
The diagram shows two circles enclosed in a rectangle. What is the distance between the centres of the circles?
Lisa's bucket weighs 21 kg when full of water. After she pours out half the water it weighs 12 kg. What is the weight of the empty bucket?
Three-quarters of the junior members of a tennis club are boys and the rest are girls. What is the ratio of boys to girls among these members?
It takes four gardeners four hours to dig four circular flower beds, each of diameter 4 metres. How long will it take six gardeners to dig six circular flower beds, each of diameter six metres?
If three runners run at the same constant speed around the race tracks, in which order do they finish?
Given the rates at which men can paint, how long will it take them to paint the Forth Bridge?
What is the relationship between the width of wide screen and traditional televisions if the area of the two screens is the same?
Three people run up stairs at different rates. If they each start from a different point - who will win, come second and come last?
A single piece of string is threaded through five holes on a piece of card. How is this possible?
A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?
If three brothers will get £20 more if they do not share their money with their sister. How much money is there?
When coins are put into piles of six 3 remain and in piles of eight 7 remain. How many remain when they are put into piles of 24?
Walking up a steep hill, I pass 10 equally spaced street lamps. How long do I take to walk from the first lamp to the last?
What is the ratio of the area of the table covered twice to the uncovered area?
The square of a number is 12 more than the number itself. The cube of the number is 9 times the number. What is the number?
A circle of radius 1 rolls without slipping round the inside of a square of side length 4. Find an expression for the number of revolutions the circle makes.
The Bean family are very particular about beans. At every meal all Beans eat some beans... At their last meal they ate 23 beans altogether. How many beans did Pa Bean eat?
The Kings of Clubs, Diamonds, Hearts and Spades, and their respective Queens, are having an arm wrestling competition.
A robot, which is initially facing North, is programmed so that each move consists of moving 5m in a straight line and then turning clockwise through an angle which increases by 10 degrees each move. How far has it travelled by the time it is first facing due East at the end of a move?
The Famous Five have been given 20 sweets as a reward for solving a tricky crime.... how many different ways can they share the sweets?
Granny has taken up deep-sea fishing! Last week, she caught a fish so big that she had to cut it into three pieces in order to weigh it. The tail weighed 9kg and the head weighed the same as the tail plus one third of the body. The body weighed as much as the head and tail together. How much did the whole fish weigh?
If two of the sides of a right-angled triangle are 5cm and 6cm long, how many possibilities are there for the length of the third side?
In the triangle PQR, the angle QPR=40degrees and the internal bisectors of the angles at Q and R meet at S, as shown. What is the size of angle QSR?
If a×b=2, b×c=24, c×a=3 and a, b and c are positive, what is the value of a+b+c?
A regular pentagon together with three sides of a regular hexagon forma cradle. What is the size of one of the angles?
In a sequence of positive integers, every term after the first two terms is the sum of the two previous terms in the sequence. If the fifth term is 2004, what is the maximum possible value of the first term?
In this addition each letter stands for a different digit, with S standing for 3. What is the value of Y×O?
Which of the following shaded regions has an area different from the other shaded regions?
One gallon of honey provides fuel for one bee to fly about 7,000,000 miles. Roughly how many bees could fly 1000 miles if they had 10 gallons of honey?
Ten stones form an arch. What is the size of the smallest angles of the trapezoidal stones?
A 1x2x3 block is placed on an 8x8 board and rolled several times.... How many squares has it occupied altogether?
Three-quarters of the area of the rectangle has been shaded. What is the length of x?
Trinni rearanges numbers on a clock face so each adjacent pair add up to a triangle number... What number did she put where 6 would usually be?
Two circles touch, what is the length of the line that is a tangentnto both circles?
One of these numbers is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?
Inspector remorse had a difficult year in 2004. Find the percentage change in the number of cases solved in 2004 compared with 2003.
Find the length along the shortest path passing through certain points on the cube.
This regular hexagon has been divided into four trapezia and one hexagon.... what is the ratio of the lengths of sides p, q and r?
Harriet Hare and Turbo Tortoise want to cross the finish line together on their 12 mile race.... What time should Harriet set off?
The diagram shows 10 identical coins which fit exactly inside a wooden frame. What is the largest number of coins that may be removed so that each remaining coin is still unable to slide.
The Queen of Spades always lies for the whole day or tells the truth for the whole day. Which of these statements can she never say?
Gill and I went to a restaurant for lunch to celebrate her birthday....We agreed to split the total cost equally. How much did I owe Gill?
When a solid cube is held up to the light, how many of the shapes shown could its shadow have?
What is the obtuse angle between the hands of a clock at 6 minutes past 8 o'clock?
A 3x8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?
Al, Berti, Chris and Di have sums of money totalling £150... What is the difference between the amount Al and Di have?
Last year, on the television programme Antiques Roadshow... work out the approximate profit.
The diagram shows a regular pentagon and regular hexagon which overlap. What is the value of x?
Counters are placed on an 8 by 8 chessboard... What fraction of the counters are on squares of the same colour as themselves?
The diagram shows two semicircular arcs... What is the diameter of the shaded region?
Calculate the ratio of areas of these squares which are inscribed inside a semi-circle and a circle.
The information display on a train shows letters by illuminating dots in a rectangular array. What fraction of the dots in this array is illuminated?
The sum of 9 consecutive positive whole numbers is 2007. What is the difference between the largest and smallest of these numbers?
In how many ways can a square be cut in half using a single straight line cut?
If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?
A male punky fish has 9 stripes and a female punky fish has 8 stripes. I count 86 stripes on the fish in my tank. What is the ratio of male fish to female fish?
The diagram shows three squares drawn on the sides of a triangle. What is the sum of the three marked angles?
Which of these graphs could be the graph showing the circumference of a circle in terms of its diameter ?
The numbers 72, 8, 24, 10, 5, 45, 36, 15 are grouped in pairs so that each pair has the same product. Which number is paired with 10?
A 30cm x 40cm page of a book includes a 2cm margin on each side... What percentage of the page is occupied by the margins?
The diagram shows a regular pentagon. Can you work out the size of the marked angle?
If p is a positive integer and q is a negative integer, which of these expressions is the greatest?
A wooden cube with edge length 12cm is cut into cubes with edge length 1cm. What is the total length of the all the edges of these centimetre cubes?
The following sequence continues indefinitely... Which of these integers is a multiple of 81?
The sum of each column and row in this grid give the totals as shown. What number goes in the starred square?
Grannie's watch gains 30 minutes every hour, whilst Grandpa's watch loses 30 minutes every hour. What is the correct time when their watches next agree?
The diagram shows two circles and four equal semi-circular arcs. The area of the inner shaded circle is 1. What is the area of the outer circle?
A triangle has been drawn inside this circle. Can you find the length of the chord it forms?
An ordinary die is placed on a horizontal table with the '1' face facing East... In which direction is the '1' face facing after this sequence of moves?
In the diagram in the question, how many squares, of any size, are there whose entries add up to an even total?
The diagram shows a semi-circle and an isosceles triangle which have equal areas. What is the value of tan x?
If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?
The flag is given a half turn anticlockwise about the point O and is then reflected in the dotted line. What is the final position of the flag?
Quince, quonce and quance are three types of fruit. Can you work out the order of heaviness of the fruits?
Malcolm and Nikki run at different speeds. They set off in opposite directions around a circular track. Where on the track will they meet?
A child's box of bricks contains cubes, cones and spheres. Can you work out how many spheres will balance a single cone?
A recipe for flapjacks calls for a certain quantity of butter, sugar and oats. Given the amount of these ingredients I have, how many flapjacks can I make?
Weighing the baby at the clinic was a problem. The baby would not keep still so we had to hold her while on the scales. Can you work out our combined weight?
A quadrilateral can have four right angles. What is the largest number of right angles an octagon can have?
How big is the angle between the hour hand and the minute hand of a clock at twenty to five?
How many pairs of numbers of the form x, 2x+1 are there in which both numbers are prime numbers less than 100?
A class raises money for charity by placing 10p pieces edge to edge in a 'silver line'. If the line was 25m long, how much money did they make?
John, Peter, Rudolf, Susie and Tony decide to set some questions for the Schools Mathematical Challenge. Can you work out how long in total they spend setting questions?
The diagram shows a regular pentagon with two of its diagonals. If all the diagonals are drawn in, into how many areas will the pentagon be divided?
How could you use this graph to work out the weight of a single sheet of paper?
An ant is crawling in a straight line when he bumps into a one centimetre cube of sugar.If he climbs over it before before continuing on his intended route, how much does the detour add to the length of his journey?
Baby can't stand up yet, so we measure her upside down. Can you use our measurements to work out how much Baby has grown in her first year?
The lengths SP, SQ and SR are equal and the angle SRQ is x degrees. What is the size of angle PQR?
This grid can be filled up using only the numbers 1, 2, 3, 4, 5 so that each number appears just once in each row, once in each column and once in each diagonal. Which number goes in the centre square?
A car with 5 tyres (four road tyres and a spare) travelled 30,000 km. All 5 tyres were used equally. How many kilometres' wear did each tyre receive?
A knitted scarf uses three balls of wool. How many balls of wool do I have at the end of the day?
Our school dinners offer the same choice each day, and each day I try a new option. How long will it be before I eat the same meal again?
The 16 by 9 rectangle is cut as shown. Rearrange the pieces to form a square. What is the perimeter of the square?
Bilbo and Frodo use a rhyme as they count their own cherry stones - where will they finish if they count all stones together?
In a triangle the smallest angle is 20 degrees. What is the largest possible angle in the triangle?
The net shown here is cut out and folded to form a cube. Which face is then opposite the face marked X?
Marcus' atrium was a square with each side 50 pedes long. How many times did Marcus have to walk round his atrium to complete his daily exercise of 8 stadia?
A car can go r miles on s gallons of petrol. How many gallons of petrol would it need for a journey of t miles?
Five friends live in five towns in southern France. In which town should they meet to keep the total travelling distance as small as possible?
Arrange these three famous mathematicians in order with the shortest-lived first.
How far away was the lightning if the flash and the thunderclap were 6 seconds apart?
When I looked at the greengrocer's window I saw a sign. When I went in and looked from the other side, what did I see?
How does Snow White need to change her value for mean-dwarf-height after a mix-up?
Each letter stands for a different digit, and S is non-zero. Which letter has the lowest value?
The perimeter of a large triangle is 24 cm. What is the total length of the black lines used to draw the figure?
How much does u litres of unleaded cost in terms of the cost per litre of 4 star petrol?
M is the midpoint of the side of the rectangle. What is the area (in square units) of the triangle PMR?
Fill in the grid with A-E like a normal Su Doku. Which letter is in the starred square?
Think of any whole number. Each time you perform a sequence of operations on it, what do you notice about the divisors of your answer?
I have a max/min thermometer in my greenhouse. What were the maximum and minimum temperatures recorded when I looked at it on Wednesday evening?
In a village the pub, church and school are at different lengths and bearings from each other. What is the bearing of the school from the church?
I am standing behind five pupils who are signalling a five-digit number to someone on the opposite side of the playground. What number is actually being signalled?
Two equilateral triangles have been drawn on two adjacent sides of a square. What is the angle between the triangles?
From the mean weight of five ballet dancers and the mean weight of ten rugby, what is the average weight of all fifteen people?
How many different ways could we have sat on the two remaining musical chairs at Gill's fourth birthday party?
From the mean of 64 numbers, and the mean of the first 36 of these numbers, can you work out the mean of the last 28 numbers?
Sam is holding two lengths of rope, and Pat ties two of the loose ends at random. What ropes could they end up with, and with what probability?
This sequence is given by the mean of the previous two terms. What is the fifth term in the sequence?
At a cinema a child's ticket costs £$4.20$ and an adult's ticket costs £$7.70$. How much did is cost this group of adults and children to see a film?
Beatrix has a 24-hour digital clock on a glass table. How many times in a 24-hour period will the display and its reflection give the same time?
This diagram has symmetry of order four. Can you use different geometric properties to find a particular length?
A garden has the shape of a right-angled triangle. A fence goes from the corner with the right-angle to a point on the opposite side. How long is the fence?
Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?
The English mathematician Augustus de Morgan has given his age in algebraic terms. Can you work out when he was born?
Dan and Ann have 9 and 8 coins respectively. What is the smallest number of coins they must swap so they end up with equal amounts of money.
Tom and Jerry start with identical sheets of paper. Each one cuts his sheet in a different way. Can you find the perimeter of the original sheet?
Mark writes four points on a line at different lengths. What is the distance between the two points furthest apart?
Gar the Magician plays a card trick on his friends Kan and Roo. Can you work out his trick and find out the sum on Kan's cards?
If this class contains between $45$% and $50$% girls, what is the smallest possible number of girls in the class?
Mr Ross tells truths or lies depending on the day of the week. Can you catch him out?
Heidi and Peter are walking through the mountains. They pass two signs which say how far away their destination is, so can you work out how long it will take them to get there?
Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?
Frank and Gabriel competed in a 200m race. Interpret the different units used for their times to work out who won.
Try to calculate the length of this diagonal line. Are you able to find more than one method?
Can Emily increase her average test score to more than $80$%? Find out how many more tests she must take to do so.
What is the largest number of digits that could be erased from this 1000-digit number, to get a surprising result?
Can you find a pair of numbers such that their sum, product and quotient are all equal? Are there any other pairs?
Is it wise for Jane to use this certain method for choosing her padlock code? Try to work out all possible combinations she might use.
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?
Three faces of a $3x3$ cube are painted red, and the other three are painted blue. How many of the 27 smaller cubes have at least one red and at least one blue face?
Three circles of different radii each touch the other two. What can you deduce about the arc length between these points?
These numbers have been written as percentages. Can you work out which has the greatest value?
This pattern repeats every 12 dots. Can you work out what a later piece will be?
An isosceles triangle is drawn inside another triangle. Can you work out the length of its base?
Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.
Mary is driving to Birmingham Airport. Using her average speed for the entire journey, find how long her journey took.
For what numbers are each of these statements true? How many of the statements can be true at the same time?
How many positive integers $n$ exist for which $n^2$ has the same number of digits as $n^3$?
Can you work out the fraction of the larger square that is covered by the shaded area?
Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?
An ant is crawling around the edges of a cube. From the description of his path, can you predict when he will return to his starting point?
The product of four different positive integers is 100. What is the sum of these four integers?
If you know three lengths and an angle in this diagram, can you find another angle by calculation?
Can you remove the least number of points from this diagram, so no three of the remaining points are in a straight line?
Can you find the solution to this equation? Each of the different letters stands for a different number.
The squares of this grid contain one of the letters P, Q, R and S. Can you complete this square so that touching squares do not contain the same letter? How many possibilities are there?
Extend two of the sides of a nonagon to form an angle. How large is this acute angle?
This pattern is made from small shaded squares. Can you picture where the patterns lead? How many squares will you need for the tenth pattern?
Consider a 10-digit number which contains only the numbers 1, 2 or 3. How many such numbers can you write so that every pair of adjacent digits differs by 1?
Two fractions have been placed on a number line. Where should another fraction be placed?
Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?
When I place a triangle over a small square, or cover a larger square with the same triangle, a certain proportion of each is covered. What is the area of the triangle?
Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area?
Leonard writes down a sequence of numbers. Can you find a formula to predict the seventh number in his sequence?
Use facts about the angle bisectors of this triangle to work out another internal angle.
Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?
Roo wants to puts stickers on the cuboid he has made from little cubes. Will he have any stickers left over?
This grocer wants to arrange his fruit in a particular order, can you help him?
Barbara is putting draughts on a $4x4$ board in a particular way. Can you find the least number of draughts she needs to put down?
Kanga hops ten times in one of four directions. At how many different points can he end up?
Andrea has just filled up a fraction of her car's petrol tank. How much petrol does she now have?
Every third page number in this book has been omitted. Can you work out what number will be on the last page?
Draw an equilateral triangle onto one side of a square. Can you work out one particular angle?
Tom and Tim are travelling towards Glasgow, but leave at different times. If Tim overtakes Tom, how fast is he travelling?
Sarah's average speed for a journey was 2 mph, and her return average speed was 4 mph. What is her average speed for the whole journey?
Can you find the time between 3 o'clock and 10 o'clock when my digital clock looks the same from both the front and back?
Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?
Pizza, Indian or Chinese takeaway. Each teenager from a class only likes two of these, but can you work which two?
Look at the times that Harry, Christine and Betty take to pack boxes when working in pairs, to find how fast Christine can pack boxes by herself.
From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?
Can you find a number and its double using the digits $1$ to $9$ only once each?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Lauren and Thomas tell their ages in terms of sums of squares. Can you work out how old they really are?
