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This is part of our collection of Short Problems.
You may also be interested in our longer problems on Reasoning, Convincing and Proving.
Here are collections of problems about Reasoning, Convincing and Proof in geometric contexts.
A collection of short problems on Angles, Polygons and Geometrical Proof.
A collection of short problems on Pythagoras's Theorem and Trigonometry.
Here are problems about Reasoning, Convincing and Proof in a variety of other contexts.
What is the smallest number of jelly babies Tom must take, to be certain that he gets at least one of each flavour?
Weekly Problem 21 - 2010
How many diagonals can you draw on this square...
Can you work out the values of J, M and C in this sum?
Jack does a 20-question quiz. How many questions didn't he attempt?
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 8 - 2016
Can you work out the size of the angles in a quadrilateral?
Fill in the grid with A-E like a Sudoku. Which letter is in the starred square?
The 30 students in a class have 25 different birthdays between them. What is the largest number that can share any birthday?
Four cards have a number on one side and a phrase on the back. On each card, the number does not have the property described on the back. What do the four cards have on them?
Weekly Problem 39 - 2016
In the diagram, VWX and XYZ are congruent equilateral triangles. What is the size of angle VWY?
Mr Ross tells truths or lies depending on the day of the week. Can you catch him out?
Weekly Problem 32 - 2015
Can you work out the missing numbers in this multiplication magic square?
Weekly Problem 38 - 2017
In the diagram, what is the value of $x$?
Is it possible to arrange the numbers 1-6 on the nodes of this diagram, so that all the sums between numbers on adjacent nodes are different?
If Clara spends £23 on books and magazines, how many of each does she buy?
What is the largest number Sophie can use to have seven positive integers with a mean of 7?
Each digit of a positive integer is 1, 2 or 3, and each of these occurs at least twice. What is the smallest such integer that is not divisible by 2 or 3?
If it takes 852 digits to number all the pages of a book, what is the number of the last page?
Weekly Problem 31 - 2017
The triangle HIJ has the same area as the square FGHI. What is the distance from J to the line extended through F and G?
You may have met Magic Squares, now meet an Anti-Magic Square. Its properties are slightly different - can you still solve it?
In this addition each letter stands for a different digit, with S standing for 3. What is the value of YxO?
Can you place the nine cards onto a 3x3 grid such that every row, column and diagonal has a product of 1?
Can you work out the number of chairs at a cafe from the number of legs?
This grid can be filled so that each of the numbers 1, 2, 3, 4, 5 appears just once in each row, column and diagonal. Which number goes in the centre square?
Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?
Is it possible to arrange the numbers 1-6 on the nodes of this diagram, so that all the sums between numbers on adjacent nodes are different?
If an athlete takes 10 minutes longer to walk, run and cycle three miles than he does to cycle all three miles, how long does it take him?
How many bees could fly 1000 miles if they had 10 gallons of honey?
Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?
The 80 spokes of The London Eye are made from 4 miles of cable. What is the approximate circumference of the wheel?
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 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?
Jasmine buys three different types of plant. How many triffids did she buy?
In a supermarket, there are two lines of tightly packed trolleys. What is the length of one trolley?
Pizza, Indian or Chinese takeaway? If everyone liked at least one, how many only liked Indian?
Weekly Problem 41 - 2016
The diagram shows a square, with lines drawn from its centre. What is the shaded area?
One of N coins is slightly heavier than the others. How large can N be if the coin can be determined with only two weighings with a set of scales?
What is the largest number of intersection points that a triangle and a quadrilateral can have?
When the numbers from 1 to 1000 are written on a blackboard, which digit appears the most number of times?
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?
This problem challenges you to find cubic equations which satisfy different conditions.