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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.

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

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Problem Description

Here's a program  illustrating an efficient way to exactly cover a given rectangle reasonable shaped rectangles.

Background & Techniques

"Robot Rooms" implements an algorithm for exactly covering a rectangular area with random rectangles meeting certain size and shape constraints. The authors' 2001 paper "Data Set
Generation for Rectangular Placement Problems", C. L. Valenzuela and P. Y. Wang, is available at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.3218

The paper slightly complex but well presented and requires no
more than high school math and a few hours of study to digest. It proves the existence of exact rectangle coverings with rectangles that are constrained by aspect ratio (are not too long and narrow) and by area ratio (restricts range in size). The proofs provide the basis for an included  algorithm for generating random coverings.

It was just what a long time DFF viewer needed generate a random arrangement of rectangles (rooms) for a Delphi investigation of intelligent robot behavior. I helped him by providing a way to create "doorways" connecting adjacent rooms. I found the problem so interesting that I independently implemented the "room generator" algorithm as well.

The program allows user control of base size, aspect and area ratios, and number of rooms to create. Two features  used for debugging were retained : 1.) an option to display text for generated rectangles and 2.) the random seed used to generate each set of rooms. This can be useful for generating the same set of rooms multiple times.

Notes for programmers

The paper referenced above proceeds step by step proving several theorems working up to:

1. Theorems 1 through 4: Given an aspect ratio, ρ (Greek letter Rho),  representing  H/W of a rectangle, and a starting rectangle whose aspect ratio is between 1/ρ and ρ, it is possible to subdivide the starting rectangle into an arbitrary number of smaller rectangles, each of which meet the same aspect ratio criteria.

2.Theorem 5: Given an area ratio. ϒ (Greek letter Upsilon), representing largest area divided by smallest area of a set of rectangles, it is possible to subdivide any starting rectangle  into an arbitrary number of smaller rectangles, each set of which meet the area ratio criteria.

3. Theorem 6:  Theorems 1 through 5  can  apply to a starting rectangle meeting the aspect ration initial condition. This allows arbitrary subdivision into as many rectangles as desired, each meeting both Rho and Upsilon conditions and cumulatively exactly covering the original rectangle.

The algorithm for implementing Theorem 6 :

 How to choose the rectangle to split next which will maintain the area ratio condition , The direction (or directions) in which to divide the rectangle (horizontally or vertically), and  the range of offsets from the top or left end of an edge which will maintain both the aspect and area ratio conditions..

Although the procedure described is recursive in nature, (each rectangle has the same steps applied) it seemed natural to implement the solution as a loop:

While more rectangles are needed
Select a rectangle to split
 Get area, m,  of largest rectangle Make a list of rectangles whose area is > 2m/ρ Randomly select one.
Split it.
 If selected rectangle can be split either Horizontally or Vertically, randomly choose one.  Otherwise just use the only valid direction. For split direction chosen, calculate the valid range of offsets. Select random split point within this range to define a new right or bottom edge. Replace existing rectangle definition with left or top  rectangle and create a new rectangle defining the right or bottom portion the split.
End

Everything else, as they say, is details.

The "door generator" portion of the program makes vertical doorways by sorting the rectangles by increasing left edge X offsets within top edge Y offsets and then checks top and bottom edge lines with the following rectangles looking for vertical edges which are co-linear and overlap. Given two colinear edges, function Overlap returns offsets of the overlapping portion.  These overlap line segments  are candidates for receiving a  doorway if they are wider than the specified door width. The same process, sorting by increasing Y coordinates within left edge offsets will find candidates for horizontal doors.

Suggestions for Further Explorations

 Original Date: January 10, 2012 Modified: July 29, 2017