Weekly Problem Archive

Stage: 3 Challenge Level:
The weekly problems are short UKMT tasks aimed at the KS3 level. They can be used as a springboard into NRICH problems.

Weekly Problem 36 - 2012

Weekly Problem 36 - 2012

Weekly Problem 37 - 2012

Weekly Problem 37 - 2012

Weekly Problem 38 - 2012

Weekly Problem 38 - 2012

Weekly Problem 39 - 2012

Weekly Problem 39 - 2012

Weekly Problem 40 - 2012

Weekly Problem 40 - 2012

Weekly Problem 36 - 2011

Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?

Weekly Problem 37 - 2011

Rotating a pencil twice about two different points gives surprising results...

Weekly Problem 38 - 2011

Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?

Weekly Problem 39 - 2011

Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radii.

Weekly Problem 40 - 2011

You may have seen magic squares before, but can you work out the missing numbers on this magic star?

Weekly Problem 41 - 2011

This magic square has only been partially completed. Can you still solve it...

Weekly Problem 42 - 2011

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?

Weekly Problem 43 - 2011

The Queen of Hearts has lost her tarts! She asks each knave if he has eaten them, but how many of them are honest...

Weekly Problem 44 - 2011

You have already used Magic Squares, now meet an Anti-Magic Square. Its properties are slightly different, but can you still solve it...

Weekly Problem 45 - 2011

What shapes can be made by folding an A4 sheet of paper only once?

Weekly Problem 46 - 2011

Multiply a sequence of n terms together. Can you work out when this product is equal to an integer?

Weekly Problem 47 - 2011

Place equal, regular pentagons together to form a ring. How many pentagons will be needed?

Weekly Problem 48 - 2011

Do these powers look odd...

Weekly Problem 49 - 2011

How long will it take Inspector Remorse to crack this crime spree?

Weekly Problem 50 - 2011

Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?

Weekly Problem 51 - 2011

The equation $x^2+2=y^3$ looks nearly quadratic. What integer solutions can you find?

Weekly Problem 52 - 2011

Draw two intersecting rectangles on a sheet of paper. How many regions are enclosed? Can you find the largest number of regions possible?

Weekly Problem 53 - 2011

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?

Weekly Problem 1 - 2012

Weekly Problem 1 - 2012

Weekly Problem 3 - 2012

Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.

Weekly Problem 4 - 2012

What fraction of the volume of this can is filled with lemonade?

Weekly Problem 5 - 2012

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Weekly Problem 6 - 2012

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Weekly Problem 8 - 2013

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Weekly Problem 8 - 2012

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Weekly Problem 9 - 2012

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Weekly Problem 10 - 2012

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Weekly Problem 11- 2012

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Weekly Problem 12 - 2012

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Weekly Problem 13 - 2012

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Weekly Problem 14 - 2012

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Weekly Problem 15 - 2012

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Weekly Problem 18 - 2012

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Weekly Problem 26 - 2012

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Weekly Problem 27 - 2012

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Weekly Problem 28 - 2012

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Weekly Problem 29 - 2012

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Weekly Problem 30 - 2012

Weekly Problem 30 - 2012

Weekly Problem 16 - 2012

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Weekly Problem 17 - 2012

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Weekly Problem 25 - 2012

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Weekly Problem 31 - 2012

Weekly Problem 31 - 2012

Weekly Problem 18 - 2013

Weekly Problem 18 - 2013

Weekly Problem 53 - 2012

ABCDEFGHI is a regular nine-sided polygon (called a 'nonagon' or 'enneagon'). What is the size of the angle FAE ?

Weekly Problem 32 - 2012

Weekly Problem 32 - 2012

Weekly Problem 19 - 2013

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Weekly Problem 20 - 2013

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Weekly Problem 22 - 2013

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Weekly Problem 15 - 2013

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Weekly Problem 21 - 2013

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Weekly Problem 35 - 2012

Weekly Problem 35 - 2012

Weekly Problem 17 - 2013

Weekly Problem 17-2013

Weekly Problem 34 - 2012

Weekly Problem 34 - 2012

Weekly Problem 13 - 2013

Weekly Problem 13-2013

Weekly Problem 52 - 2012

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?

Weekly Problem 14 - 2013

Weekly Problem 14-2013

Weekly Problem 2 - 2013

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Weekly Problem 16 - 2013

Weekly Problem 16-2013

Weekly Problem 51 - 2012

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?

Weekly Problem 33 - 2012

Weekly Problem 33 - 2012

Weekly Problem 3 - 2013

Weekly Problem 3-2013

Weekly Problem 50 - 2012

The diagram shows a regular dodecagon. What is the size of the marked angle?

Weekly Problem 23 - 2013

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Weekly Problem 24 - 2013

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Weekly Problem 27 - 2013

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Weekly Problem 28 - 2013

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Weekly Problem 32 - 2013

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Weekly Problem 25 - 2013

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Weekly Problem 26 - 2013

Weekly problem - 2013

Weekly Problem 29 - 2013

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Weekly Problem 30 - 2013

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Weekly Problem 34 - 2013

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Weekly Problem 33 - 2013

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Weekly Problem 35 - 2013

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Weekly Problem 41 - 2013

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Weekly Problem 40 - 2013

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Weekly Problem 36 - 2013

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Weekly Problem 37 - 2013

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Weekly Problem 45 - 2013

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Weekly Problem 48 - 2013

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Weekly Problem 49 - 2013

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Weekly Problem 50 - 2013

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Weekly Problem 51 - 2013

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weekly problem 52 - 2013

Weekly Problem 1 - 2006

The diagram shows two circles enclosed in a rectangle. What is the distance between the centres of the circles?

Weekly Problem 2 - 2006

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?

Weekly Problem 3 - 2006

weekly problem 3-2006

Weekly Problem 4 - 2006

weekly problem 4-2006 Work out the radius of a roll of adhesive tape.

Weekly Problem 5 - 2006

weekly problem 5-2006

Weekly Problem 6 - 2006

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?

Weekly Problem 7 - 2006

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?

Weekly Problem 8 - 2006

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Weekly Problem 9 - 2006

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Weekly Problem 10 - 2006

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Weekly Problem 11 - 2006

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Weekly Problem 12 - 2006

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Weekly Problem 13 - 2006

If three runners run at the same constant speed around the race tracks, in which order do they finish?

Weekly Problem 14 - 2006

Given the rates at which men can paint, how long will it take them to paint the Forth Bridge?

Weekly Problem 15 - 2006

What is the relationship between the width of wide screen and traditional televisions if the area of the two screens is the same?

Weekly Problem 16 - 2006

Three people run up stairs at different rates. If they each start from a different point - who will win, come second and come last?

Weekly Problem 17 - 2006

Granny says she is 84 years old, not counting her Sundays. How old is she?

Weekly Problem 19 - 2006

What is the total area enclosed by the three semicicles?

Weekly Problem 18 - 2006

What is the area of the pentagon?

Weekly Problem 20 - 2006

A single piece of string is threaded through five holes on a piece of card. How is this possible?

Weekly Problem 21 - 2006

How high is this table?

Weekly Problem 22 - 2006

A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?

Weekly Problem 23 - 2006

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Weekly Problem 28 - 2006

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Weekly Problem 29 - 2006

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Weekly Problem 30 - 2006

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Weekly Problem 32 - 2006

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Weekly Problem 33 - 2006

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Weekly Problem 36 - 2006

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Weekly Problem 41 - 2006

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Weekly Problem 34 - 2006

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Weekly Problem 35 - 2006

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Weekly Problem 37 - 2006

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Weekly Problem 40 - 2006

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Weekly Problem 43 - 2006

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Weekly Problem 48 - 2006

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Weekly Problem 50 - 2006

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Weekly Problem 51 - 2006

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Weekly problem 52 - 2006

Weekly Problem 1 - 2007

Three integers have a sum of 1 and a product of 36 - what are they?

Weekly Problem 6 - 2007

Who is the youngest in this family?

Weekly Problem 2 - 2007

How many different flags can you make?

Weekly Problem 7 - 2007

How many rats did the Pied Piper catch?

Weekly Problem 4 - 2007

How much more antifreeze is needed to make the proportion 30%?

Weekly Problem 8 - 2007

If three brothers will get Â£20 more if they do not share their money with their sister. How much money is there?

Weekly Problem 5 - 2007

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?

Weekly Problem 9 - 2007

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?

Weekly Problem 3 - 2007

What is the ratio of the area of the table covered twice to the uncovered area?

Weekly Problem 10 - 2007

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?

Weekly Problem 11 - 2007

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.

Weekly Problem 13 - 2007

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?

Weekly Problem 15 - 2007

The Kings of Clubs, Diamonds, Hearts and Spades, and their respective Queens, are having an arm wrestling competition.

Weekly Problem 16 - 2007

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?

Weekly Problem 22 - 2007

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?

Weekly Problem 21 - 2007

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?

Weekly Problem 23 - 2007

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?

Weekly Problem 14 - 2007

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?

Weekly Problem 17 - 2007

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?

Weekly Problem 18 - 2007

A regular pentagon together with three sides of a regular hexagon forma cradle. What is the size of one of the angles?

Weekly Problem 44 - 2006

Weekly Problem 23 - 2007

Weekly Problem 19 - 2007

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?

Weekly Problem 20 - 2007

In this addition each letter stands for a different digit, with S standing for 3. What is the value of Y×O?

Weekly Problem 43 - 2006

To make porridge, Goldilocks mixes oats and wheat bran..... what percentage of the mix is wheat?

Weekly Problem 24 - 2007

Which of the following shaded regions has an area different from the other shaded regions?

Weekly Problem 25 - 2007

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?

Weekly Problem 26 - 2007

The diagram shows two equilateral triangles. What is the value of x?

Weekly Problem 27 - 2007

Ten stones form an arch. What is the size of the smallest angles of the trapezoidal stones?

Weekly Problem 28 - 2007

A 1x2x3 block is placed on an 8x8 board and rolled several times.... How many squares has it occupied altogether?

Weekly Problem 29 - 2007

Weekly Problem 29 - 2007

Weekly Problem 30 - 2007

Three-quarters of the area of the rectangle has been shaded. What is the length of x?

Weekly Problem 31 - 2007

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?

Weekly Problem 33 - 2007

Two circles touch, what is the length of the line that is a tangent to both circles?

Weekly Problem 32 - 2007

One of these numbers is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?

Weekly Problem 34 - 2007

Inspector remorse had a difficult year in 2004. Find the percentage change in the number of cases solved in 2004 compared with 2003.

Weekly Problem 35 - 2007

What is the area of the triangle formed by these three lines?

Weekly Problem 36 - 2007

Find the length along the shortest path passing through certain points on the cube.

Weekly Problem 37 - 2007

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?

Weekly Problem 38 - 2007

Find the missing number if the mean, median and mode are all the same.

Weekly Problem 39 - 2007

A solid 'star' shape is created. How many faces does it have?

Weekly Problem 53 - 2009

Sydney flew to Melbourne, Australia. What time was it in Melbourne when Sydney's flight arrived?

Weekly Problem 40 - 2007

Harriet Hare and Turbo Tortoise want to cross the finish line together on their 12 mile race.... What time should Harriet set off?

Weekly Problem 43 - 2007

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.

Weekly Problem 41 - 2007

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?

Weekly Problem 44 - 2007

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?

Weekly Problem 46 - 2007

When a solid cube is held up to the light, how many of the shapes shown could its shadow have?

Weekly Problem 45 - 2007

What is the obtuse angle between the hands of a clock at 6 minutes past 8 o'clock?

Weekly Problem 47 - 2007

What is 50% of 2007 plus 2007% of 50?

Weekly Problem 48 - 2007

A 3x8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?

Weekly Problem 49 - 2007

What is the mean of 1.2 recurring and 2.1 recurring?

Weekly Problem 50 - 2007

Al, Berti, Chris and Di have sums of money totalling Â£150... What is the difference between the amount Al and Di have?

Weekly Problem 51 - 2007

Last year, on the television programme Antiques Roadshow... work out the approximate profit.

Weekly Problem 52 - 2007

Can you work out the value of x in this 'power-full' equation?

Weekly Problem 53 - 2007

The diagram shows a regular pentagon and regular hexagon which overlap. What is the value of x?

Weekly Problem 1 - 2008

Given that the number 2008 is the correct answer to a sum, can you find n?

Weekly Problem 3 - 2008

Counters are placed on an 8 by 8 chessboard... What fraction of the counters are on squares of the same colour as themselves?

Weekly Problem 2 - 2008

The diagram shows two semicircular arcs... What is the diameter of the shaded region?

Weekly Problem 5 - 2008

Calculate the ratio of areas of these squares which are inscribed inside a semi-circle and a circle.

Weekly Problem 4 - 2008

In the figure given in the problem, calculate the length of an edge.

Weekly Problem 6 - 2008

From this sum of powers, can you find the sum of the indices?

Weekly Problem 7 - 2008

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?

Weekly Problem 9 - 2008

The sum of 9 consecutive positive whole numbers is 2007. What is the difference between the largest and smallest of these numbers?

Weekly Problem 8 - 2008

In how many ways can a square be cut in half using a single straight line cut?

Weekly Problem 10 - 2008

If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?

Weekly Problem 12 - 2008

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?

Weekly Problem 11 - 2008

The mean of three numbers, x, y and z is x. What is the mean of y and z?

Weekly Problem 13 - 2008

The diagram shows three squares drawn on the sides of a triangle. What is the sum of the three marked angles?

Weekly Problem 15 - 2008

Which of these graphs could be the graph showing the circumference of a circle in terms of its diameter ?

Weekly Problem 14 - 2008

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?

Weekly Problem 16 - 2008

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?

Weekly Problem 18 - 2008

The diagram shows a regular pentagon. Can you work out the size of the marked angle?

Weekly Problem 17 - 2008

If p is a positive integer and q is a negative integer, which of these expressions is the greatest?

Weekly Problem 19 - 2008

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?

Weekly Problem 22 - 2008

The following sequence continues indefinitely... Which of these integers is a multiple of 81?

Weekly Problem 21 - 2008

The sum of each column and row in this grid give the totals as shown. What number goes in the starred square?

Weekly Problem 20 - 2008

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?

Weekly Problem 24 - 2008

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?

Weekly Problem 23 - 2008

A triangle has been drawn inside this circle. Can you find the length of the chord it forms?

Weekly Problem 25 - 2008

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?

Weekly Problem 27 - 2008

In the diagram in the question, how many squares, of any size, are there whose entries add up to an even total?

Weekly Problem 28 - 2008

The diagram shows a semi-circle and an isosceles triangle which have equal areas. What is the value of tan x?

Weekly Problem 26 - 2008

If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?

Weekly Problem 30 - 2008

How many minutes are there between 11.41 and 14.02?

Weekly Problem 31 - 2008

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?

Weekly Problem 33 - 2008

Quince, quonce and quance are three types of fruit. Can you work out the order of heaviness of the fruits?

Weekly Problem 32 - 2008

Malcolm and Nikki run at different speeds. They set off in opposite directions around a circular track. Where on the track will they meet?

Weekly Problem 35 - 2008

A child's box of bricks contains cubes, cones and spheres. Can you work out how many spheres will balance a single cone?

Weekly Problem 37 - 2008

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?

Weekly Problem 36 - 2008

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?

Weekly Problem 38 - 2008

A quadrilateral can have four right angles. What is the largest number of right angles an octagon can have?

Weekly Problem 39 - 2008

How big is the angle between the hour hand and the minute hand of a clock at twenty to five?

Weekly Problem 41 - 2008

How many pairs of numbers of the form x, 2x+1 are there in which both numbers are prime numbers less than 100?

Weekly Problem 40 - 2008

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?

Weekly Problem 34 - 2008

What is the area of the region common to this triangle and square?

Weekly Problem 44 - 2008

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?

Weekly Problem 45 - 2008

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?

Weekly Problem 46 - 2008

How could you use this graph to work out the weight of a single sheet of paper?

Weekly Problem 47 - 2008

Can you solve this magic square?

Weekly Problem 48 - 2008

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?

Weekly Problem 49 - 2008

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?

Weekly Problem 50 - 2008

The lengths SP, SQ and SR are equal and the angle SRQ is x degrees. What is the size of angle PQR?

Weekly Problem 51 - 2008

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?

Weekly Problem 52 - 2008

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?

Weekly Problem 42 - 2007

How many triangles have all three angles perfect squares (in degrees)?

Weekly Problem 43 - 2008

A knitted scarf uses three balls of wool. How many balls of wool do I have at the end of the day?

Weekly Problem 1 - 2009

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?

Weekly Problem 2 - 2009

The 16 by 9 rectangle is cut as shown. Rearrange the pieces to form a square. What is the perimeter of the square?

Weekly Problem 4 - 2009

Bilbo and Frodo use a rhyme as they count their own cherry stones - where will they finish if they count all stones together?

Weekly Problem 3 - 2009

What fraction of the area of this regular hexagon is the shaded triangle?

Weekly Problem 6 - 2009

In a triangle the smallest angle is 20 degrees. What is the largest possible angle in the triangle?

Weekly Problem 7 - 2009

Tony and Tina can't work out which of them owes what to the other. Can you?

Weekly Problem 5 - 2009

The net shown here is cut out and folded to form a cube. Which face is then opposite the face marked X?

Weekly Problem 9 - 2009

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?

Weekly Problem 8 - 2009

How many ways can Gill, the baby, arrange the letters in her name?

Weekly Problem 10 - 2009

In how many ways can you give change for a ten pence piece?

Weekly Problem 11 - 2009

How many of the numbers 1 to 20 are not the sum of two primes?

Weekly Problem 12 - 2009

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?

Weekly Problem 13 - 2009

How many zeros does 50! have at the end?

Weekly Problem 14 - 2009

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?

Weekly Problem 16 - 2009

Arrange these three famous mathematicians in order with the shortest-lived first.

Weekly Problem 15 - 2009

Can you work out Ali's age based on the diagram?

Weekly Problem 17 - 2009

How far apart are the first and last bus stops in this bus route?

Weekly Problem 18 - 2009

How much lighter will Â£5 worth of 5p's be with these new lighter coins?

Weekly Problem 20 - 2009

How far away was the lightning if the flash and the thunderclap were 6 seconds apart?

Weekly Problem 19 - 2009

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?

Weekly Problem 21 - 2009

What is the angle between the two hands of a clock at 2.30?

Weekly Problem 22 - 2009

How much money did the Queen give away in pence as a power of 2?

Weekly Problem 23 - 2009

How does Snow White need to change her value for mean-dwarf-height after a mix-up?

Weekly Problem 24 - 2009

Each letter stands for a different digit, and S is non-zero. Which letter has the lowest value?

Weekly Problem 29 - 2008

The seven pieces in this 12 cm by 12 cm square make a Tangram set. What is the area of the shaded parallelogram?

Weekly Problem 25 - 2009

Jack does a 20-question quiz. How many questions didn't he attempt?

Weekly Problem 27 - 2009

The perimeter of a large triangle is 24 cm. What is the total length of the black lines used to draw the figure?

Weekly Problem 28 - 2009

How much does u litres of unleaded cost in terms of the cost per litre of 4 star petrol?

Weekly Problem 30 - 2009

M is the midpoint of the side of the rectangle. What is the area (in square units) of the triangle PMR?

Weekly Problem 29 - 2009

Fill in the grid with A-E like a normal Su Doku. Which letter is in the starred square?

Weekly Problem 31 - 2009

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?

Weekly Problem 32 - 2009

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?

Weekly Problem 33 - 2009

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?

Weekly Problem 34 - 2009

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?

Weekly Problem 35 - 2009

Two equilateral triangles have been drawn on two adjacent sides of a square. What is the angle between the triangles?

Weekly Problem 26 - 2009

From the mean weight of five ballet dancers and the mean weight of ten rugby, what is the average weight of all fifteen people?

Weekly Problem 42 - 2008

How many different ways could we have sat on the two remaining musical chairs at Gill's fourth birthday party?

Weekly Problem 36 - 2009

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?

Weekly Problem 37 - 2009

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?

Weekly Problem 38 - 2009

This sequence is given by the mean of the previous two terms. What is the fifth term in the sequence?

Weekly Problem 39 - 2009

Can you show this algebraic expression is divisible by 4?

Weekly Problem 40 - 2009

This quadrilateral has an unusual shape. Are you able to find its area?

Weekly Problem 41 - 2009

At a cinema a child's ticket costs Â£$4.20$ and an adult's ticket costs Â£$7.70$. How much did is cost this group of adults and children to see a film?

Weekly Problem 42 - 2009

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?

Weekly Problem 43 - 2009

This diagram has symmetry of order four. Can you use different geometric properties to find a particular length?

Weekly Problem 44 - 2009

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?

Weekly Problem 45 - 2009

Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?

Weekly Problem 46 - 2009

How many squares can you draw on this lattice?

Weekly Problem 47 - 2009

The English mathematician Augustus de Morgan has given his age in algebraic terms. Can you work out when he was born?

Weekly Problem 48 - 2009

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.

Weekly Problem 49 - 2009

Weekly Problem 49 - 2009

Weekly Problem 50 - 2009

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?

Weekly Problem 51 - 2009

Mark writes four points on a line at different lengths. What is the distance between the two points furthest apart?

Weekly Problem 52 - 2009

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?

Weekly Problem 1 - 2010

How many cubes can you see...

Weekly Problem 2 - 2010

Can you solve this 'KANGAROO' alphanumeric subtraction?

Weekly Problem 3 - 2010

If this class contains between $45$% and $50$% girls, what is the smallest possible number of girls in the class?

Weekly Problem 4 - 2010

Mr Ross tells truths or lies depending on the day of the week. Can you catch him out?

Weekly Problem 5 - 2010

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?

Weekly Problem 6 - 2010

Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?

Weekly Problem 7 - 2010

Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

Weekly Problem 8 - 2010

Are you able to find triangles such that these five statements are true?

Weekly Problem 9 - 2010

Frank and Gabriel competed in a 200m race. Interpret the different units used for their times to work out who won.

Weekly Problem 10 - 2010

Try to calculate the length of this diagonal line. Are you able to find more than one method?

Weekly Problem 11 - 2010

Can you DISCOVER the smallest solution to this word problem...

Weekly Problem 12 - 2010

Can Emily increase her average test score to more than $80$%? Find out how many more tests she must take to do so.

Weekly Problem 13 - 2010

The Seven Dwarfs have seven different birthdays. How old can they be?

Weekly Problem 14 - 2010

What is the largest number of digits that could be erased from this 1000-digit number, to get a surprising result?

Weekly Problem 15 - 2010

Can you find a pair of numbers such that their sum, product and quotient are all equal? Are there any other pairs?

Weekly Problem 16 - 2010

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.

Weekly Problem 17 - 2010

The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?

Weekly Problem 18 - 2010

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?

Weekly Problem 19 - 2010

Three circles of different radii each touch the other two. What can you deduce about the arc length between these points?

Weekly Problem 20 - 2010

You have already used Magic Squares, now meet a Magic Octahedron...

Weekly Problem 21 - 2010

How many diagonals can you draw on this square...

Weekly Problem 22 - 2010

Can you form this 2010-digit number...

Weekly Problem 23 - 2010

These numbers have been written as percentages. Can you work out which has the greatest value?

Weekly Problem 24 - 2010

Can you find a number that is half way between these two fractions?

Weekly Problem 25 - 2010

These four touching circles have another circle hiding amongst them...

Weekly Problem 26 - 2010

Does Joseph have too many sheep in his flock...

Weekly Problem 28 - 2010

Manipulate one algebraic fraction to solve another.

Weekly Problem 27 - 2010

This pattern repeats every 12 dots. Can you work out what a later piece will be?

Weekly Problem 29 - 2010

An isosceles triangle is drawn inside another triangle. Can you work out the length of its base?

Weekly Problem 30 - 2010

Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.

Weekly Problem 31 - 2010

Mary is driving to Birmingham Airport. Using her average speed for the entire journey, find how long her journey took.

Weekly Problem 32 - 2010

For what numbers are each of these statements true? How many of the statements can be true at the same time?

Weekly Problem 33 - 2010

How many positive integers $n$ exist for which $n^2$ has the same number of digits as $n^3$?

Weekly Problem 34 - 2010

Can you work out the fraction of the larger square that is covered by the shaded area?

Weekly Problem 35 - 2010

Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?

Weekly Problem 36 - 2010

How many squares are needed to continue this pattern?

Weekly Problem 37 - 2010

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?

Weekly Problem 12 - 2007

Weekly problem 12 - 2007

Weekly Problem 38 - 2010

The product of four different positive integers is 100. What is the sum of these four integers?

Weekly Problem 39 - 2010

If you know three lengths and an angle in this diagram, can you find another angle by calculation?

Weekly Problem 40 - 2010

Can you remove the least number of points from this diagram, so no three of the remaining points are in a straight line?

Weekly Problem 41 - 2010

Can you make this solid with six triangular faces "Magic"?

Weekly Problem 42 - 2010

Can you find the solution to this equation? Each of the different letters stands for a different number.

Weekly Problem 43 - 2010

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?

Weekly Problem 44 - 2010

Extend two of the sides of a nonagon to form an angle. How large is this acute angle?

Weekly Problem 45 - 2010

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?

Weekly Problem 46 - 2010

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?

Weekly Problem 47 - 2010

Two fractions have been placed on a number line. Where should another fraction be placed?

Weekly Problem 48 - 2010

Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?

Weekly Problem 49 - 2010

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?

Weekly Problem 50 - 2010

Find a simple way to compute this long fraction.

Weekly Problem 51 - 2010

Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area?

Weekly Problem 52 - 2010

Leonard writes down a sequence of numbers. Can you find a formula to predict the seventh number in his sequence?

Weekly Problem 1 - 2011

Use facts about the angle bisectors of this triangle to work out another internal angle.

Weekly Problem 2 - 2011

Using the new operator $\oplus$, can you solve this equation?

Weekly Problem 3 - 2011

What does Pythagoras' Theorem tell you about the radius of these circles?

Weekly Problem 4 - 2011

10 must remain within easy reach...

Weekly Problem 5 - 2011

Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

Weekly Problem 6 - 2011

Powers of numbers might look large, but which of these is the largest...

Weekly Problem 7 - 2011

Roo wants to puts stickers on the cuboid he has made from little cubes. Will he have any stickers left over?

Weekly Problem 8 - 2011

This grocer wants to arrange his fruit in a particular order, can you help him?

Weekly Problem 9 - 2011

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?

Weekly Problem 10 - 2011

Will this product give a perfect square?

Weekly Problem 11 - 2011

Kanga hops ten times in one of four directions. At how many different points can he end up?

Weekly Problem 12 - 2011

How many numbers do you need to remove to avoid making a perfect square?

Weekly Problem 13 - 2011

The sum of three square numbers equals $121$. What can those numbers be...

Weekly Problem 14 - 2011

What does the sum of these three numbers tell us about their product?

Weekly Problem 15 - 2011

Andrea has just filled up a fraction of her car's petrol tank. How much petrol does she now have?

Weekly Problem 16 - 2011

How does the perimeter change when we fold this isosceles triangle in half?

Weekly Problem 17 - 2011

Every third page number in this book has been omitted. Can you work out what number will be on the last page?

Weekly Problem 18 - 2011

Draw an equilateral triangle onto one side of a square. Can you work out one particular angle?

Weekly Problem 19 - 2011

Tom and Tim are travelling towards Glasgow, but leave at different times. If Tim overtakes Tom, how fast is he travelling?

Weekly Problem 20 - 2011

What is the perimeter of this unusually shaped polygon...

Weekly Problem 21 - 2011

How many ways can you paint this wall with four different colours?

Weekly Problem 22 - 2011

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?

Weekly Problem 23 - 2011

MatildaMatildaMatil... What is the 1000th letter?

Weekly Problem 24 - 2011

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?

Weekly Problem 25 - 2011

Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?

Weekly Problem 26 - 2011

Alberta won't reveal her age. Can you work it out from these clues?

Weekly Problem 27 - 2011

Pizza, Indian or Chinese takeaway. Each teenager from a class only likes two of these, but can you work which two?

Weekly Problem 28 - 2011

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.

Weekly Problem 29 - 2011

From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

Weekly Problem 30 - 2011

Three touching circles have an interesting area between them...

Weekly Problem 31 - 2011

Can you find a number and its double using the digits $1$ to $9$ only once each?

Weekly Problem 32 - 2011

What could be the scores from five throws of this dice?

Weekly Problem 33 - 2011

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Weekly Problem 34 - 2011

Lauren and Thomas tell their ages in terms of sums of squares. Can you work out how old they really are?

Weekly Problem 35 - 2011

Weekly Problem 35

Weekly Problem 9 - 2013

Weekly Problem 9 - 2013

Weekly Problem Archive

An archive of the weekly problems, which are UKMT question at the KS3 level.

Weekly Problems

Hundreds of weekly problems for your enjoyment

Weekly Problem 2 - 2012

Weekly Problem 2 - 2012

Weekly Problem 1 - 2013

Weekly Problem 1-2013

Weekly Problem 49 - 2006

Weekly problem 49 - 2006

Weekly Problem 41 - 2012

Weekly Problem 41 - 2012

Weekly Problem 42 - 2012

Weekly Problem 42 - 2012

Weekly Problem 43 - 2012

Weekly Problem 43 - 2012

Weekly Problem 44 - 2012

Consider two arithmetic sequences: 1998, 2005, 2012,... and 1996, 2005, 2014,... Which numbers will appear in both?

Weekly Problem 45 - 2012

There are 10 girls in a mixed class. If two pupils are selected, the probability that they are both girls is 0.15. How many boys are in the class?

Weekly Problem 46 - 2012

If a number is even, halve it; if odd, treble it and add 1. If a sequence starts at 13, what will be the value of the 2008th term?

Weekly Problem 47 - 2012

Weekly Problem 47 - 2012

Weekly Problem 48 - 2012

The curve $y=x^2âˆ’6x+11$ is rotated through $180^\circ$ about the origin. What is the equation of the new curve?

Weekly Problem 49 - 2012

Sketch the graph of the curve $y^2 = x(2âˆ’x)$

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