Rectangle Counts

[Home]   [Puzzles & Projects]    [Delphi Techniques]   [Math topics]   [Library]   [Utilities]

 

Search

Search WWW

Search DelphiForFun.org

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  They do say "Offer may be withdrawn at any time", so don't delay if you want to check it out.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.

 

Support DFF - Shop

 If you shop at Amazon anyway,  consider using this link. We receive a few cents from each purchase.   Thanks.


Support DFF - Donate

 If you benefit from the website,  in terms of knowledge, entertainment value, or something otherwise useful, consider making a donation via PayPal  to help defray the costs.  (No PayPal account necessary to donate via credit card.)  Transaction is secure.

Contact

Feedback:  Send an e-mail with your comments about this program (or anything else).

Search DelphiForFun.org only

 

 

Here's a puzzle sent to me by a viewer who says it is from a set of questions written by computer pioneer and inventor Sir Clive Sinclair. They were published many years ago as "Mensa Steps" in a magazine, perhaps "Design Technology" as the viewer recalls ,

===========================
"If you draw a nine by nine square, thus giving yourself eighty-one small squares in total;  how many rectangles can you count in total? "
===========================

One of my standard problem solving strategies is to "simplify"; solve a smaller version of the program and hope that it provides a clue to solving the larger one. The program  generates the number of rectangles for grids from from 1x1 to 9x9. It does it in a nested loop by summing all rectangle counts from 1x1 to NxN for each grid size.  We count rectangles of size H x V where  H and V represent horizontal and vertical dimensions and range from  1 to N. For each H there are  H+1-A rectangles across and for each V there are N+1-V  rectangles down.  The total numer of rectangles is the sum of all products (N+1-H)*(N+1-V) for all H and V combinations from 1 to N.

I did not discover the generating function based solely on the results, but if you search on the sequences of results (1,9,36,100,225,441 ...) you'll find lots of fascinating information including the formula in the Online Encyclopedia of Integer Sequences page at  http://oeis.org/A000537. Also notice that the numbers in the sequence are perfect squares of the sequence 1,3,6,10,15.... These are called "triangular numbers" with the property that the Nth member is the number of dots in an equilateral triangle with N dots per side! (Think of  4th entry for example, being  represented by 10 bowling pins arranged in the traditional triangle with 4 pins per side.)  See http://en.wikipedia.org/wiki/Squared_triangular_number for more interesting information.

I was planning to post this program on the Beginner's page since there are less than 20 lines of code, but decided on the Math Topics section after recognizing  the interesting math.

 Running/Exploring the Program 

 

Created March 6,,2012

Modified February 18, 2016

 

 

  [Feedback]   [Newsletters (subscribe/view)] [About me]
Copyright 2000-2017, Gary Darby    All rights reserved.