Maths Puzzles


The following puzzles are from the Transum monthly newsletters and podcasts.

Multiple Remainders

What is the second smallest number such that when it is divided by 5 the remainder is 4, and when divided by 7 the remainder is 6?

Coaches to Cambridge

Can you figure out which coach I travelled in when I went to Cambridge?

Percy Cod's Kids

Percy Cod was talking about his children. Can you work out their ages given the ratios?

Carrot-eating Critters

Work out how many of each kind of animal are in the field from their carrot-eating habits.

Red Arrow Square

What fraction of the square does the red arrow cover?

Patsy Loves Percy

The strange case of Percy who seems to be getting older very quickly

Monkeys, Kittens and Dogs

Who is most likely to be able to work out the square root of 121?

Feed The Horses

How long will the remaining feed last the horses that have not been sold?

Strange Addition

If I start at five and add six I get eleven but if I start at six and add seven I get one.

Prime Permutations

Of all the permutations of 1 to 9 used to make nine-digit numbers, how many are prime?

4-Digit Hotel Room Safe Code

Can you work out what number I used to lock and unlock my hotel room safe?

Choir Eye Colour

Figure out the percentage of choir members that do not have blue eyes from the clues given.

Calculator Keys at the Corners of a Rectangle

A question about the four keys at the corners of a rectangle on a calculator.

Average House Numbers

Work out the house numbers from the clues about the mean, median and mode.

Clock in the Mirror

What time was the clock in the mirror really showing?

Delightfully Divisible

Find a pandigital number that is delightfully divisible.

Five integers with a product of 12

Can you find five integers that multiply together to give twelve?

Square in a Rectangle

What is the largest square that can be drawn in the corner of a 10cm by 15cm rectangle?

London Marathon

A question about the average speed required for the second half of the marathon.

Cutting the Lawn

Aynuk and Ayli cut half of the lawn each. How long is each side of the square lawn?

The Missing Pound's Found

A really wonderful answer to the missing pound puzzle.

Forty Five in Four Parts

Split the number 45 into four parts according to the given information.

Shrivelled Spuds

Work out the weight of the potatoes after they have been left out in the sun to dry.

Transposition Error

Work out the bank balance given information about the transposition error.

Counting Sheep

Work out the number of sheep owned by Percy and Patsy from the given clues.

Juggling with Egg Timers

Can you time exactly nine minutes using the four and seven minute egg timers?

Permutation Sum

What is sum of all the four digit numbers containing all of the digits one to four?

Odd Probability

What is the probability that two random numbers are both even if they are not both odd?

Regions in Circles

Calculate the number of regions in a circle formed by intersecting chords.

Brothers and Sisters

Can you work out the number of children in the Numlove family given the clues about brothers and sisters?

Exam Average

What mark is required in the last exam to achieve an 80% overall average?

Cake Cut

Where should the last slice of cake be cut to give two equal pieces?

Holy Sphere

Calculate the remaining volume of the sphere after a cylindrical hole has been drilled through the centre

Jumping Flea

How many different places could the flea find itself after 8 foot-long jumps either north, south, east or west?

Last Digit

How many positive two-digit numbers are there whose square and cube both end in the same digit?

Central Station

The probability that the next train to leave will be going north is five times the probability that the next train to leave will be going south.

Canteen Queue

Is it possible to answer the question if Betsy's age is not known?

Separated Twins

Work out the combination of the safe given the clues about pairs of numbers.

Divisible By Three

A puzzle about two digit numbers that can be made from ten different digits.

Letters In Numbers

A brand new puzzle involving the letters in numbers when written as words.

Square Angled Triangle

The angles of a triangle are all square numbers. What are they?

Tri-Junction Puzzle

What is the probability of the three cars arriving at the road junction not being involved in an accident?

Two Prime Squares

What is the smallest square number (greater than one) that cannot be expressed as the sum of two prime numbers?

The Missing Pound

Where did the missing pound go in this story about three people visiting a restaurant?

Ticks, Tocks, Tacks and Tucks

Find how ticks compare to tocks, tacks and tucks from the given information.

The Power of Christmas

A question about indices to get you thinking mathematically at this festive time of year.

Ant and Dec

What single question could Dec ask Ant to find out what he is thinking?

Bookworm

How far does the bookworm travel on its meal-time journey?

Three Mathematicians

How can the third mathematician be so certain that everyone wants a drink?

Unfinished Game

If the coin tossing game was cut short how would you share the winnings?

Alphanumbetical

Questions about the letters used to make the natural numbers.

Best Dice

Which of the unusual dice would you choose to give you the best chance of winning the prize?

The Birthday Problem

What is the probability of two or more pupils in a class having the same birthday?

Enigma Vowels

A coding question inspired by the film The Imitation Game.

Biased Coin

A mathematical question asked by Microsoft and Google interviewers.

Twelve Days of Christmas

Can you figure out exactly how many gifts the true love sends during the twelve days of the Christmas holiday?

Chased by a Bear

A puzzle about an explorer being chased by a bear along with a question about imperial and metric measures with Measurement Man

Halloween Bases

A puzzle about Why is halloween like Christmas along with news of the new Number Skills Inventory

Torch Tunnel

A puzzle about four people making their way through a tunnel with just one torch along with news of the new numerology page

Cube in Milk

A puzzle about a cube being lowered into a bucket of milk along with news of the new shunting puzzles


The following puzzles are the Transum Advanced Starters.

Algebraic Product

Finding the value of the expression is easier than you think!

Angle Thinking

Find the range of possible angles, x, for which tan x > cos x > sin x

Average Cycling Speed

Work out the average speed of two journeys. The obvious answer is not the correct answer.

Back To The Factory

Find all the numbers below 1000 which have exactly 20 factors

Barmy BIDMAS

A misleading way of stating the answer to a simple calculation.

Best Dice

Which of the unusual dice would you choose to give you the best chance of winning the prize?

Biased Coin

Use a biased coin to obtain a fair result

Cheryl's Birthday

Use a process of elimination to work out the correct date from the clues given.

Coordinate Distance

Find k given that(-2,k) is 13 units away from (10,9)

Cuboid

Find the dimensions of a cuboid matching the description given

Difference Cipher

Find the mathematical word from the cipher

Divisible by 11

Can you prove that a three digit number whose first and third digits add up to the value of the second digit must be divisible by eleven?

Double or Half?

At ten percent change per day is doubling achieved faster than halving?

Exceeds by 99

Find the number whose double exceeds its half by exactly 99.

Fence Optimisation

Find the length of a rectangle enclosing the largest possible area.

Fermi Problem

The classic Fermi problem using standard estimation techniques

Find The Radius

Find the radius of the circle from the small amount of information provided.

Four Fraction Division

Explain why the answer to a series of fraction divisions is a whole number.

GDC Challenge

Produce the given graph on a graphic display calculator

Geometry Snack

Find the value of the marked angle in this diagram from the book Geometry Snacks

Grandmother

How far would grandma have travelled after a suitably large number of days given her walking regime?

Hands Together

The hands of a clock are together at midnight. At what time are they next together?

HCF and LCM given

If given the HCF, LCM and the smaller of two numbers can you find the other?

How Many Left Handers?

Work out the number of members if the probability of left-handed members being randomly selected is given.

Hundred and Fifty Percent

Divide 110 into two parts so that the larger part is 150% of the smaller part.

Key Eleven

Prove that a four digit number constructed in a certain way will be a multiple of eleven.

Log Perfection

Determine whether the given statements containing logarithms are true or false

Logarithm Equation

Solve an equation containing logarithms with different bases

Maximum Product

Two numbers add up to 10. What's the largest possible product they could have?

Multiple of sum of digits

What number is six times the sum of its digits?

Nine Digit Numbers

How many different nine digit numbers are their that contain each of the digits from one to nine?

Other Child's Gender

What is the probability that the other child is also a boy?

Paper Ratio

Calculate the ratio of the sides of an A4 sheet of paper without any measuring.

Paper Surprising Perimeter

Find the perimeter of a folded sheet of A4 paper as described in this short video.

Parallel Graphs

Determine from their equations which of the straight line graphs are parallel and perpendicular.

Penny Bags

Can you place 63 pennies in bags in such a way that you can give away any amount of money (from 1p to 63p) by giving a selection of these prepacked bags?

Permutable Functions

Find pairs of functions that are commutative under composition.

Piece of String

Find where a piece of string should be cut to form a circle and a square of equal areas.

Pizza Slice

A problem which can be solved by considereing the areas of a triangle and a sector of a circle.

Quad Midpoints

What shape is created when the midpoints of the sides of a quadrilateral are joined together?

Restrained Flea

How many different places could the flea find itself after 8 foot-long jumps either north, south, east or west?

Reverse Connection

Find a general rule for the difference between a two digit number and that same number with the digits reversed.

Rice on a Chess Board

How many grains of rice are on a chess board if each square has twice the number of grains as the previous square.

Road Connections

Design roads to connect four houses that are on the corners of a square, side of length one mile, to minimise the total length of the roads.

Same Series Sum

Find an arithmetic series and a geometric series that have the same sum of the first five terms.

Same Three Digits

Find expressions which contain indices and evaluate to numbers containing the same digits.

Simplify

Simplify an algebraic fraction

Single Fraction

Simplify an expression involving fractions, exponents and a quare root.

Speed Circles

Find the diameters of the circles in the corners of the square.

Sphere Hole

Find the volume of the remaining part of a sphere after a 10cm cylindrical hole has been drilled through it.

Square in Rectangle

Find the area of a square drawn under the diagonal of a rectangle

Square Root of 1%

What is the square root of one percent?

Tan 22.5

Find an exact value for tan 22.50 without using a calculator.

Test Scores

Explore the misconception that when adding fractions you add both the numerators and the denominators

Three Right Triangles

Calculate the lengths of the unlabelled sides of these right-angled triangles.

Tri-Junction

A real life situation that can be analysed with the use of a tree diagram.

Triangle or Quadrilateral

Can a quadrilateral have a straight angle?

Two Equals One

What is wrong with the algebraic reasoning that shows that 2 = 1 ?

Two real numbers

The sum of the reciprocals of two real numbers is -1, and the sum of their cubes is 4. What are they?

Unfinished Game

Share the prize in a fair ratio according to the probability of each player willing.

Unlucky Seven Eleven

Follow the instructions to multiply a chosen number then explain the result you get.

Vowel Code

How many ways can you create a code for the vowels by assigning to each vowel a different vowel?

Weather Reports

Which five different integers multiply together to give 12?

What Question?

Write down all the possible questions that could have been asked if this was the diagram provided in a mathematics textbook.

X Divided by 2Y

Why do different calculators not agree on the order of operations?

Zero Even

Prove that zero is an even number.

Apple

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