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

### Weekly Problem 1 - 2006

### Weekly Problem 2 - 2006

### Weekly Problem 6 - 2006

### Weekly Problem 7 - 2006

### Weekly Problem 13 - 2006

### Weekly Problem 14 - 2006

### Weekly Problem 15 - 2006

### Weekly Problem 16 - 2006

### Weekly Problem 20 - 2006

### Weekly Problem 22 - 2006

### Weekly Problem 8 - 2007

### Weekly Problem 5 - 2007

### Weekly Problem 9 - 2007

### Weekly Problem 3 - 2007

### Weekly Problem 10 - 2007

### Weekly Problem 11 - 2007

### Weekly Problem 13 - 2007

### Weekly Problem 15 - 2007

### Weekly Problem 16 - 2007

### Weekly Problem 22 - 2007

### Weekly Problem 21 - 2007

### Weekly Problem 23 - 2007

### Weekly Problem 14 - 2007

### Weekly Problem 17 - 2007

### Weekly Problem 18 - 2007

### Weekly Problem 19 - 2007

### Weekly Problem 20 - 2007

### Weekly Problem 43 - 2006

### Weekly Problem 24 - 2007

### Weekly Problem 25 - 2007

### Weekly Problem 27 - 2007

### Weekly Problem 28 - 2007

### Weekly Problem 30 - 2007

### Weekly Problem 31 - 2007

### Weekly Problem 33 - 2007

### Weekly Problem 32 - 2007

### Weekly Problem 34 - 2007

### Weekly Problem 36 - 2007

### Weekly Problem 37 - 2007

### Weekly Problem 53 - 2009

### Weekly Problem 40 - 2007

### Weekly Problem 43 - 2007

### Weekly Problem 41 - 2007

### Weekly Problem 44 - 2007

### Weekly Problem 46 - 2007

### Weekly Problem 45 - 2007

### Weekly Problem 48 - 2007

### Weekly Problem 50 - 2007

### Weekly Problem 51 - 2007

### Weekly Problem 53 - 2007

### Weekly Problem 3 - 2008

### Weekly Problem 2 - 2008

### Weekly Problem 5 - 2008

### Weekly Problem 7 - 2008

### Weekly Problem 9 - 2008

### Weekly Problem 8 - 2008

### Weekly Problem 10 - 2008

### Weekly Problem 12 - 2008

### Weekly Problem 13 - 2008

### Weekly Problem 15 - 2008

### Weekly Problem 14 - 2008

### Weekly Problem 16 - 2008

### Weekly Problem 18 - 2008

### Weekly Problem 17 - 2008

### Weekly Problem 19 - 2008

### Weekly Problem 22 - 2008

### Weekly Problem 21 - 2008

### Weekly Problem 20 - 2008

### Weekly Problem 24 - 2008

### Weekly Problem 23 - 2008

### Weekly Problem 25 - 2008

### Weekly Problem 27 - 2008

### Weekly Problem 28 - 2008

### Weekly Problem 26 - 2008

### Weekly Problem 31 - 2008

### Weekly Problem 33 - 2008

### Weekly Problem 32 - 2008

### Weekly Problem 35 - 2008

### Weekly Problem 37 - 2008

### Weekly Problem 36 - 2008

### Weekly Problem 38 - 2008

### Weekly Problem 39 - 2008

### Weekly Problem 41 - 2008

### Weekly Problem 40 - 2008

### Weekly Problem 44 - 2008

### Weekly Problem 45 - 2008

### Weekly Problem 46 - 2008

### Weekly Problem 48 - 2008

### Weekly Problem 49 - 2008

### Weekly Problem 50 - 2008

### Weekly Problem 51 - 2008

### Weekly Problem 52 - 2008

### Weekly Problem 43 - 2008

### Weekly Problem 29 - 2008

### Weekly Problem 27 - 2009

### Weekly Problem 28 - 2009

### Weekly Problem 30 - 2009

### Weekly Problem 29 - 2009

### Weekly Problem 31 - 2009

### Weekly Problem 32 - 2009

### Weekly Problem 33 - 2009

### Weekly Problem 34 - 2009

### Weekly Problem 35 - 2009

### Weekly Problem 42 - 2008

### Weekly Problem 36 - 2009

### Weekly Problem 37 - 2009

### Weekly Problem 38 - 2009

### Weekly Problem 41 - 2009

### Weekly Problem 42 - 2009

### Weekly Problem 43 - 2009

### Weekly Problem 44 - 2009

### Weekly Problem 45 - 2009

### Weekly Problem 47 - 2009

### Weekly Problem 48 - 2009

### Weekly Problem 50 - 2009

### Weekly Problem 51 - 2009

### Weekly Problem 52 - 2009

### Weekly Problem 20 - 2015

### Clone of Weekly Problem 18 - 2009

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 x b=2, b x c=24, c x 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 form a 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 YxO?

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

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 tangent to 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?

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

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?

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

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?

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?

Four brothers give statements about the order they were born in. Can you work out which two are telling the truth?

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