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Have you got a Super Brain?

Adapting and exploring maths challenge problems is an excellent way of finding ideas for IB maths explorations and extended essays.  This problem is taken from the book: The first 25 years of the Superbrain challenges.  I’m going to see how many different ways I can solve it.

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The problem is to find all the integer solutions to the equation above.  Finding only integer solutions is a fundamental part of number theory – a branch of mathematics that only deals with integers.  

Method number 1: Brute force

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This is a problem that computers can make short work of.  Above I wrote a very simple Python program which checked all values of x and y between -99 and 99.  This returned the only solution pairs as:

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Clearly we have not proved these are the only solutions – but even by modifying the code to check more numbers, no more pairs were found.

Method number 2: Solving a linear equation

We can notice that the equation is linear in terms of y, and so rearrange to make y the subject.

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We can then use either polynomial long division or the method of partial fractions to rewrite this.  I’ll use partial fractions.  The general form for this fraction can be written as follows:

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Next I multiply by the denominator and the compare coefficients of terms.

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This therefore gives:

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I can now see that there will only be an integer solution for y when the denominator of the fraction is a factor of 6.  This then gives (ignoring non integer solutions):

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I can then substitute these back to find my y values, which give me the same 4 coordinate pairs as before:

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Method number 3: Solving a quadratic equation

I start by making a quadratic in x:

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I can then use the quadratic formula to find solutions:

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Which I can simplify to give:

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Next I can note that x will only be an integer solution if the expression inside the square root is a square number.  Therefore I have:

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Next I can solve a new quadratic as follows:

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As before I notice that the expression inside my square root must be a square number.  Now I can see that I need to find m and n such that I have 2 square numbers with a difference of 24.  I can look at the first 13 square numbers to see that from the 12th and 13th square numbers onwards there will also be a difference of more than 24.  Checking this list I can find that m = 1 and m = 5 will satisfy this equation.

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This then gives:

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which when I solve for integer solutions and then sub back into find x gives the same four solutions:

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Method number 4: Graphical understanding

Without rearranging I could imagine this as a 3D problem by plotting the 2 equations:

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This gives the following graph:

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We can see that the plane intersects the curve in infinite places.  I’ve marked A, B on the graph to illustrate 2 of the coordinate pairs which we have found.  This is a nice visualization but doesn’t help find our coordinates, so lets switch to 2D.

In 2D we can use our rearranged equation:

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This gives the following graph:

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Here I have marked on the solution pairs that we found.   The oblique asymptote (red) is y = 2x-1 because as x gets large the fraction gets very small and so the graph gets closer and closer to y = 2x -1. 

All points on this curve are solutions to the equation – but we can see that the only integer solution pairs will be when x is small.  When x is a large integer then the curve will be close to the asymptote and hence will return a number slightly bigger than an integer.

So, using this approach we would check all possible integer solutions when x is small, and again should be able to arrive at our coordinate pairs.

So, 4 different approaches that would be able to solve this problem.  Can you find any others?

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.  

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.