Other tags that relate to Base Puzzle
Number theory. Diophantine equations. Multiplication & division. Creating and manipulating expressions and formulae. Place value. Number bases. Factors and multiples. Integers. Mathematical reasoning & proof. Addition & subtraction.There are 160 results
Broad Topics > Thinking Mathematically > Mathematical reasoning & proof
Tis Unique
Age 11 to 14
Challenge Level
This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.
Aba
Age 11 to 14
Challenge Level
In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.
Never Prime
Age 14 to 16
Challenge Level
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
Number Rules - OK
Age 14 to 16
Challenge Level
Can you produce convincing arguments that a selection of statements about numbers are true?
Gabriel's Problem
Age 11 to 14
Challenge Level
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Euler's Squares
Age 14 to 16
Challenge Level
Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
Diophantine N-tuples
Age 14 to 16
Challenge Level
Can you explain why a sequence of operations always gives you perfect squares?
Our Ages
Age 14 to 16
Challenge Level
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
Composite Notions
Age 14 to 16
Challenge Level
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
Adding All Nine
Age 11 to 14
Challenge Level
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .
Postage
Age 14 to 16
Challenge Level
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
More Mathematical Mysteries
Age 11 to 14
Challenge Level
Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .
Power Mad!
Age 11 to 14
Challenge Level
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Common Divisor
Age 14 to 16
Challenge Level
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
The Genie in the Jar
Age 11 to 14
Challenge Level
This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .
Largest Product
Age 11 to 14
Challenge Level
Which set of numbers that add to 10 have the largest product?
Perfectly Square
Age 14 to 16
Challenge Level
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Chocolate Maths
Age 11 to 14
Challenge Level
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
One O Five
Age 11 to 14
Challenge Level
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
A Biggy
Age 14 to 16
Challenge Level
Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.
Even So
Age 11 to 14
Challenge Level
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
Why 24?
Age 14 to 16
Challenge Level
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
Geometric Parabola
Age 14 to 16
Challenge Level
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
Always the Same
Age 11 to 14
Challenge Level
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
What Numbers Can We Make Now?
Age 11 to 14
Challenge Level
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Mod 3
Age 14 to 16
Challenge Level
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Archimedes and Numerical Roots
Age 14 to 16
Challenge Level
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
Thirty Nine, Seventy Five
Age 11 to 14
Challenge Level
We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .
Ordered Sums
Age 14 to 16
Challenge Level
Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .
Janine's Conjecture
Age 14 to 16
Challenge Level
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
Is it Magic or Is it Maths?
Age 11 to 14
Challenge Level
Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .
Calendar Capers
Age 11 to 14
Challenge Level
Choose any three by three square of dates on a calendar page...
N000ughty Thoughts
Age 14 to 16
Challenge Level
How many noughts are at the end of these giant numbers?
Always Perfect
Age 14 to 18
Challenge Level
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
DOTS Division
Age 14 to 16
Challenge Level
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
Cross-country Race
Age 14 to 16
Challenge Level
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
Top-heavy Pyramids
Age 11 to 14
Challenge Level
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Chameleons
Age 11 to 14
Challenge Level
Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12. . . .
Find the Fake
Age 14 to 16
Challenge Level
There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
What Numbers Can We Make?
Age 11 to 14
Challenge Level
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Triangular Intersection
Age 14 to 16 Short
Challenge Level
What is the largest number of intersection points that a triangle and a quadrilateral can have?
Mindreader
Age 11 to 14
Challenge Level
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Marbles
Age 11 to 14
Challenge Level
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
9 Weights
Age 11 to 14
Challenge Level
You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?
Sticky Numbers
Age 11 to 14
Challenge Level
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
Multiplication Square
Age 14 to 16
Challenge Level
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
The Great Weights Puzzle
Age 14 to 16
Challenge Level
You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?
Greetings
Age 11 to 14
Challenge Level
From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .
NRICH is part of the family of activities in the Millennium Mathematics Project.