Pythagorean Triples

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Mensa Daily Puzzlers

For over 15 years Mensa Page-A-Day calendars have provided several puzzles a year for my programming pleasure.  Coding "solvers" is most fun, but many programs also allow user solving, convenient for "fill in the blanks" type.  Below are Amazon  links to the two most recent years.

Mensa 365 Puzzlers  Calendar 2017

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(Hint: If you can wait, current year calendars are usually on sale in January.)


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Pythagorean Triples


Problem Description

This program solve problem #1 from Clessa's PC Puzzle book: "... Find the smallest area Pythagorean triangle whose perimeter is a perfect square and whose area is a perfect cube."  

The solution itself is not very interesting, but there are a few things that may be new for beginners.  The heart of the program is a nested loop that increments variables a and b and for each pair applies  3 tests

bulleta2 + b2  = cfor some integer c (i.e. it is a Pythagorean triple)
bullet(a + b + c) = n2 for some integer n (i.e. perimeter is a perfect square)
bullet1/2 (a*b) = n3 for some integer n (i.e. area is a perfect cube)

As soon as any test is failed, we can continue with the next b value.  If all three tests are passed, we have a solution.

If you're a beginner, there are couple of things worth noting here, you'll see them lots: 


The statements Screen.cursor:=crHourGlass and Screen.Cursor:=crDefault to set he cursor to "busy" while solving and reset it when finished. 


The "trunc" function converts real numbers to integer types. 

It is frequently desirable (or necessary) to measure the time that some task performs.  I put a couple of timing tests in here just to see how they work.  I originally used the "Now" function which gets date and time values but it's not very accurate for measurements under a second  (solving the program takes 6 milliseconds (.oo6 seconds) on my computer.   I won't go into the detail of date/time formats here, suffice it to say that Now usually returned 0 ms with an occaisonal  50 ms result thrown in -  kind of useless for timing.   Two more accurate functions used here are "timeGetTime" and "QueryPerformanceCounter".  These are calls to Windows functions and are described in the "Windows SDK"  Help file.  (SDK stands for System Development Kit, by the way.)    

As always, feel free to email me if you have problems or questions.


Running/Exploring the Program 

To view the code, click here.

To download the source code, click here.

To download the executable code, click here.

Suggestions for Further Study

bulletI wonder what the second smallest triangle is? 
bulletThere's  some commented code in the program that tries to use a "generating function" to generate the triples for testing without trying all combinations.  I couldn't get it to generate the required solution, but I think I've seen several web sites that claim that it generates all triples.  Do a search on "Pythagorean triples", and you'll find lots of links.   Anyway, since the long way of solving the problem only took .006 seconds, I guess I can't justify spending too much time looking for a faster algorithm.
bulletReplace the SQR (square) functions with multiplies (e.g. replace Sqr(a) with a*a, etc.)  Does it affect the run time?  Can you find any other ways to speed up the program?


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