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# Reasoning, Convincing and Proving - Short Problems

### Angles, Polygons and Geometrical Proof Short Problems

### Pythagoras' Theorem and Trigonometry - Short Problems

### Quiz Questions

### Kept Apart

### Other Side

### Out of Line

### Birthday Party

### What's on the Back?

### Equilateral Pair

### Weekly Lies

### Multiplication Magic Square

### Shared Vertex

### Would You Like a Jelly Baby?

### Distinct Diagonals

### Down and Along

### Sevens

### Ones, Twos and Threes

### Digital Book

### Leaning Over

### Anti-magic Square

### So Many Sums

### Magical Products

### A Leg to Stand On

### Total Totality

### Bookshop

### Knights and Knaves

### More Total Totality

### To Run or Not to Run?

### Distinct in a Line

### Hiking the Hill

### Honey Bees

### Peter's Primes

### The London Eye

### Trolley Park

### Takeaway Time

### Shaded Square

### Spot the Fake

### Long List

### Day of the Triffids

### Digital Counter

### Square LCM

### Triangular Intersection

### Curve Fitter

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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.

Age 11 to 16

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

Age 11 to 16

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.

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 8 - 2016

Can you work out the size of the angles in a quadrilateral?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 39 - 2016

In the diagram, VWX and XYZ are congruent equilateral triangles. What is the size of angle VWY?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 32 - 2015

Can you work out the missing numbers in this multiplication magic square?

Age 11 to 14

ShortChallenge Level

Weekly Problem 38 - 2017

In the diagram, what is the value of $x$?

Age 11 to 14

ShortChallenge Level

What is the smallest number of jelly babies Tom must take, to be certain that he gets at least one of each flavour?

Age 11 to 14

ShortChallenge Level

Weekly Problem 21 - 2010

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

What is the largest number Sophie can use to have seven positive integers with a mean of 7?

Age 11 to 14

ShortChallenge Level

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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

Can you place the nine cards onto a 3x3 grid such that every row, column and diagonal has a product of 1?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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?

Age 11 to 14

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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?

Age 14 to 16

ShortChallenge Level

How many bees could fly 1000 miles if they had 10 gallons of honey?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

The 80 spokes of The London Eye are made from 4 miles of cable. What is the approximate circumference of the wheel?

Age 14 to 16

ShortChallenge Level

In a supermarket, there are two lines of tightly packed trolleys. What is the length of one trolley?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

Weekly Problem 41 - 2016

The diagram shows a square, with lines drawn from its centre. What is the shaded area?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

Weekly Problem 47 - 2017

How many numbers do I need in a list to have two squares, two primes and two cubes?

Age 14 to 16

ShortChallenge Level

Jasmine buys three different types of plant. How many triffids did she buy?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Age 14 to 18

ShortChallenge Level

This problem challenges you to find cubic equations which satisfy different conditions.