A Weighty Problem (#1)

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Problem Description

Once some kids wanted to set up a lemonade stand to make money. To set themselves apart from the other lemonade stands on the block, they decided to sell their lemonade by the pound. They found an old balance beam scale, the kind with a weight pan on each side, and three weights. They discovered that they could sell any whole number of pounds from 1 to 13. What weights did they have.?

Background & Techniques

This program was motivated by an email I sent recently to my 13 year old granddaughter at camp.  (Email  at camp?   What a different would we live in today!)  

 She happened also to be in cabin 13 so in addition to the lemonade problem, I included a few facts about the number 13 designed to help overcome  triskaidekaphobia.  e.g. 

  • 13 is the smallest "emirp", a prime that forms a different prime when its digits are reversed. I wonder what the second smallest is. 
  •  Unless I'm mistaken, ELEVEN + TWO anagrams to TWELVE + ONE 
  •  The longest recorded flight of a chicken is 13 seconds. 
  • Of course 13 is also a Fibonacci number, so you can check local pine cones for 13 spirals in the seed pattern

The lemonade weighing problem is not too difficult to solve by trial and error, but it aroused my curiosity about  weight sets with 2, 4, 5, or more weights.  How many values can be weighed by each of those sets?    How can we determine the optimum values for the weights?.

There are actually two programs here - the main form searches for the best set of weights for a user specified number of weights from 1 through 5.   "Best" means capable of weighing any unknown item with integral weights  between one and the sum of all the weights.    A "Play with the scale" button on the main form displays a screen with a simple animation of a balance beam scale and  unknown weights that you can "weigh" by placing the known weights on the balance pans with the unknown item.  

After the program was written I found that there are simple analytical solutions that confirm the program results, but it was an interesting mental exercise anyway.     Here's a link to an good analytical treatment by Spyros Potamianos from newsgroup rec.puzzles and archived by Arlet  at rec-puzzles.org.  I'm not sure who Arlet is, all I could find at the site was a picture of this kind of ugly guy, but he is doing a good job of preserving the contents of an interesting newsgroup.    Search on "optimal weights" to locate the article by Spyros. 

Non-programmers are welcome to read on, but may want to skip to the bottom of this page to download an executable version of the program.

Notes for Programmers

The search procedure is a little bit complex and any description would probably sound like a journal paper;  so for the one or two people who might be interested, I've tried to put good comments in the program.  If you want to know and can't figure it out, send me an email.   

The second part, the animated scale, is easier.  A TScale object was defined as a descendent of TImage to contain the scale image and information about the weights on each pan.   Drawing is done on Tscale's Canvas with separate methods for drawing

  • the base,  DrawBase;
  • the beam which sets on top of the base and tilts 12 degrees left of right depending on which side is heavier DrawBeam;  and 
  • the pans, one suspended from each end of the beam and which visually holds the weights,  DrawPans.  

 Procedure DrawScale calls each of the above routines in succession specifying the new angle of the beam.  Drawing is performed 10 times in a loop, moving 1/10 of the total rotation angle each time through.  

Keeping track of the weights was made a little trickier by my decision to place the unknown weight on the outside of its pan and known weights on the inside.  This made it necessary to keep track of the unknown weight separately from the TList structures, LeftWeights and RightWeights,   which are used to keep track of the known weights.   

The weights are TStaticText objects just because they seemed the simplest controls that allowed internal text to be aligned and could be colored.   Since they will be dragged onto of off of the scale, they remain parented by the form.    Detecting when weights were removed was a particular problem.  OnStartDrag exits for the weights weren't always triggered for some reason.  I solved the problem by always attempting to remove a weight when it was dropped on the scale (to handle the case of a drag from a pan to a pan).  An OnDragDropExit for the Form detects the case where a weight was removed from a pan back to the form area outside the scale image. 

A few other items that might be on interest:

  • TScale uses a callback procedure (OnBalanced) to notify the form when the scale is in balance.  The  definition for a procedure type that is a method  (mine is), mist include the phrase  "of object" . So the definition is Onbalanced: Procedure of object;
  • TLists are an excellent way to keep track of objects - remember that object references are actually pointers to the objects, so space usage is minimal and detecting whether a particular object is in the list (i.e. in the weight pan) is a simple matter.
  • Drag/drop processing   basically requires three steps: 
    • change Dragmode from dmManual to dmAutomatic in the object to be dragged.
    • define a OnDragOver exit for any control where the object may be dropped.  Set the passed variable Accept to true in this exit.  The object being dropped is passed in variable Source, so you can check if necessary to make sure that you want to accept it.  
    • define an OnDragDrop exit to process the logic when the object is actually dropped.  Again the object being dropped is passed in variable Source so in our.Scale dragdrop exit we can update it position to move it to the closest weight pan, and we can add to to our list of weights in this pan.
  • In generating a random weight, I decided to also generate a random color.   There is a Windows SDK function RGB for  this, but I noticed a significant delay when using it.   My equivalent used here is unknownweight.color:=(64+random(192)) shl 16+ (64+random(192)) shl 8 + (64+random(192));  This sets each of the low order bytes of unknownweight.color to a random number between 64 and 255.  The lower limit of 64 eliminates very dark colors.  The shift left (shl) instructions could be replaced with multiply by 256 (shl 8) and 65538 (shl 16).  Delphi is smart enough to use shl to multiply by powers powers of 2.  

OK, let's get on with it!

Running/Exploring the Program 

Suggestions for Further Explorations

There's a great  On-Line Encyclopedia of Integer Sequences sponsored by AT&T.  Our sequence, for the maximum weighable item for N known weights, searchable from the index page, is number  A003462.   You can learn more about it there, and get sidetracked to many other sequences in the process.  
Also the  rec-puzzles.org archive link will provide many hours of happy browsing. 
The other frequently encountered weighing problem is the Counterfeit weight problem.  Given N coins with one of them counterfeit, and a balance beam scale, what is the minimum number of weighing necessary to find the bad coin?   In different variations we may know that the weight of the counterfeit is light, heavy, or just different.    The scale in this program could provide a starting point for a program handling problems of this type.


Original Date: July 20, 2002

Modified: February 18, 2016

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