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As of October, 2016, Embarcadero is offering a free release of Delphi (Delphi 10.1 Berlin Starter Edition ).     There are a few restrictions, but it is a welcome step toward making more programmers aware of the joys of Delphi.  They do say "Offer may be withdrawn at any time", so don't delay if you want to check it out.  Please use the feedback link to let me know if the link stops working.


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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

Mensa 365 Puzzlers Calendar 2018

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


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In the X4X2 program, we proved empirically that  X4-X2 is always a multiple of 12.  Here's a slightly more rigorous proof.  

1. Factoring out  X2 :   X4-X2=X2(X2-1)

2. Factor X2-1:    X4-X2=X2(X-1)(X+1)

3a. The expression must be divisible by 4: If X is even then X2 is divisible by 4 and the expression is divisible by 4.  If X is odd then (X-1) and (X+1) are even and again the expression is divisible by 4.  Therefore the expression is divisible by 4.

3b.  The expression must be divisible by 3:  The expression has 3 consecutive factors X-1, X, X+1.  Given any 3 consecutive integers, one of the them is divisible by 3.   Proof left as an exercise for the reader.

 4. Since the expression is divisible by 3 and by 4 it is divisible by 12.  Q.E.D.

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