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DOTS Division
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}.
Sixational
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.
Age 14 to 16
Challenge Level
Why do this problem?
This problem helps students to appreciate the power of algebra for solving problems involving number and place value. It would make a good follow-up problem for students who have worked on Reversals.
Possible approach
This printable worksheet may be useful: Puzzling Place Value
You may wish to use this problem for consolidation in class or homework, after working on Reversals or Always a Multiple.
When students have had a go at tackling the questions, take some time to discuss the answers as a class, focussing particularly on their explanations of why there were only a limited number of solutions, or why the solutions they have found satisfied a particular condition.
Key questions
Is there an algebraic expression to represent the problem?
Are there any restrictions on A, B and C?
Possible support
Always a Multiple offers multiple representations for solving place value problems, so it might be a good problem to try before looking at this one.
Possible extension
For students who are ready to explore algebraic representations with quadratic expressions, Factorising with Multilink and Pair Products may be suitable follow-up problems.
NRICH is part of the family of activities in the Millennium Mathematics Project.