Reasoning, Justifying, Convincing and Proof - Short Problems

This is part of our collection of Short Problems.

You may also be interested in our longer problems on Reasoning, Justifying, Convincing and Proof.
Here are collections of problems about Reasoning, Justifying, Convincing and Proof in geometric contexts.

Angles, Polygons and Geometrical Proof Short Problems

Age 11 to 16

A collection of short problems on Angles, Polygons and Geometrical Proof.

Pythagoras' Theorem and Trigonometry - Short Problems

Age 11 to 16

A collection of short problems on Pythagoras's Theorem and Trigonometry.



Here are problems about Reasoning, Justifying, Convincing and Proof in a variety of other contexts.

Down and Along

Age 11 to 14 Short Challenge Level:

Can you work out the values of J, M and C in this sum?

Other Side

Age 11 to 14 Short Challenge Level:

Weekly Problem 8 - 2016
Can you work out the size of the angles in a quadrilateral?

Out of Line

Age 11 to 14 Short Challenge Level:

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

Equilateral Pair

Age 11 to 14 Short Challenge Level:

Weekly Problem 39 - 2016
In the diagram, VWX and XYZ are congruent equilateral triangles. What is the size of angle VWY?

Weekly Lies

Age 11 to 14 Short Challenge Level:

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

Birthday Party

Age 11 to 14 Short Challenge Level:

The 30 students in a class have 25 different birthdays between them. What is the largest number that can share any birthday?

Multiplication Magic Square

Age 11 to 14 Short Challenge Level:

Weekly Problem 32 - 2015
Can you work out the missing numbers in this multiplication magic square?

Old Order

Age 11 to 14 Short Challenge Level:

Who is the youngest in this family?

Kept Apart

Age 11 to 14 Short Challenge Level:

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?

Shared Vertex

Age 11 to 14 Short Challenge Level:

Weekly Problem 38 - 2017
In the diagram, what is the value of $x$?

So Many Sums

Age 11 to 14 Short Challenge Level:

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

Anti-magic Square

Age 11 to 14 Short Challenge Level:

You may have met Magic Squares, now meet an Anti-Magic Square. Its properties are slightly different - can you still solve it?

A Leg to Stand On

Age 11 to 14 Short Challenge Level:

Can you work out the number of chairs at a cafe from the number of legs?

Mini Cross-number

Age 11 to 14 Short Challenge Level:

Which digit replaces x in this crossnumber?

Total Totality

Age 11 to 14 Short Challenge Level:

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?

Digital Book

Age 11 to 14 Short Challenge Level:

If it takes 852 digits to number all the pages of a book, what is the number of the last page?

Bookshop

Age 11 to 14 Short Challenge Level:

If Clara spends £23 on books and magazines, how many of each does she buy?

Distinct in a Line

Age 11 to 14 Short Challenge Level:

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?

More Total Totality

Age 11 to 14 Short Challenge Level:

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?

Knights and Knaves

Age 11 to 14 Short Challenge Level:

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

Peter's Primes

Age 14 to 16 Short Challenge Level:

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?

Takeaway Time

Age 14 to 16 Short Challenge Level:

Pizza, Indian or Chinese takeaway? If everyone liked at least one, how many only liked Indian?

Long List

Age 14 to 16 Short Challenge Level:

Weekly Problem 47 - 2017
How many numbers do I need in a list to have two squares, two primes and two cubes?

Shaded Square

Age 14 to 16 Short Challenge Level:

Weekly Problem 41 - 2016
The diagram shows a square, with lines drawn from its centre. What is the shaded area?

Spot the Fake

Age 14 to 16 Short Challenge Level:

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?

Digital Counter

Age 14 to 16 Short Challenge Level:

When the numbers from 1 to 1000 are written on a blackboard, which digit appears the most number of times?

Square LCM

Age 14 to 16 Short Challenge Level:

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