p = sqrt(N+k) + sqrt(N-k); An Interesting Mathematical problem: Maths is Fun

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Recently, I came across an interesting mathematical problem on the internet that I thought could be generalised to provide much greater insight into how mathematics works.  The problem requires use of school level maths and I found it great fun to work on. 

(Click on the slide to see its full page image)

Equations 3, 4 and 5 provide the full solution of the problem.  Only even values of p are possible and their range is defined by eqns. 3 and 4.  Eqn.5 may then be used to calculate the corresponding values of k. 

The next slide shows the allowed values of p.

The range of p is seen to be quite small - in fact for N < 200, the number of allowed p values are less than four.  

For example, for N = 80, p(min) = 12.65 and p(max) = 17.89.  Since p is an even integer, only possible values of p are 14 and 16.  The corresponding values of k are (use eqn.5) 77.95 and 64. 

Final Word:  I find the problem to be very instructive - particularly, it demonstrates that some serious conclusions may be obtained by initial observations of the problem.  For example, by looking at the expression in eqn.1, we could establish the range of p values - later confirmed by the general analysis.  

Plotting a graph of the results is also very useful as this provides a wholesome view of the results obtained.

I hope that you enjoyed our little excursion and agree that 'Maths is Fun'.