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There are **24** NRICH Mathematical resources connected to **Diophantine equations**, you may find related items under Algebraic expressions, equations and formulae.

Problem
Primary curriculum
Secondary curriculum
### Letter Land

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### CD Heaven

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at each price?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fibs

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Lattice Points

Why are there only a few lattice points on a hyperbola and infinitely many on a parabola?

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Euclid's Algorithm II

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Euclid's Algorithm I

How can we solve equations like 13x + 29y = 42 or 2x +4y = 13 with the solutions x and y being integers? Read this article to find out.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Why Stop at Three by One

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Are You Kidding

If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Deep Roots

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

Age 11 to 14

Challenge Level

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Primary curriculum
Secondary curriculum
### In Particular

Can you find formulas giving all the solutions to 7x + 11y = 100 where x and y are integers?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Whole Numbers Only

Can you work out how many of each kind of pencil this student bought?

Age 11 to 14

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Not a Polite Question

When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...

Age 11 to 14

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Coffee

To make 11 kilograms of this blend of coffee costs £15 per kilogram. The blend uses more Brazilian, Kenyan and Mocha coffee... How many kilograms of each type of coffee are used?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Hallway Borders

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?

Age 11 to 14

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Problem
Primary curriculum
Secondary curriculum
### Double Angle Triples

Try out this geometry problem involving trigonometry and number theory

Age 16 to 18

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Problem
Primary curriculum
Secondary curriculum
### Our Ages

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Exhaustion

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Some Cubes

The sum of the cubes of two numbers is 7163. What are these numbers?

Age 16 to 18

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Primary curriculum
Secondary curriculum
### BT.. Eat Your Heart Out

If the last four digits of my phone number are placed in front of the remaining three you get one more than twice my number! What is it?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Upsetting Pitagoras

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shades of Fermat's Last Theorem

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

Age 16 to 18

Challenge Level