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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 Delphi version of a classic Water Jug problem.   One version goes like this:

Two friends who have an eight-quart jug of water wish to share it evenly. They also have two empty jars, one holding five quarts, the other three. How can they each measure exactly 4 quarts of water?

(Solution takes 7 pour operations.)

### Background & Techniques

Viewer Charles Doumar wrote the original program and introduced me to the Water Jug problem as a test of our TGraphSearch unit which incorporates the Dijkstra Shortest Path Search algorithm.  Alex Bogomolny's  Cut the Knot site has a good Water Jug problem page some of the history and theory.

Because this version is designed as a test of shortest path searching, we allow any combination of starting values with  any other achievable combination as a goal.   The solver initially knows the capacity and amount of liquid contained in each jug and is searching for a given goal configuration.  The jugs have no intermediate measure of the amount contained so pouring transfers the smaller of  (1.) the amount of water in the "From" jug and (2.) the available unused capacity of the "To" jug.  And, of course, the fewer moves the better.

Users may set up any 3 or 4 jug puzzle and try to solve it by clicking on jug images.    When you're ready to give up, as I frequently am, the "Show me" button will compute the necessary moves.

#### Programmer's Notes

The program uses procedure Dijkstra from the TGraphSearch class to search for solutions.    At the top of the form the user selects the number of jugs, 3 or 4, the capacity for each jug and the total amount of water to be distributed across the jugs.

When any of these amounts change, a call to JarBtnClick  generates nodes for all possible ways to distribute the specified water amount across the available jugs.    The TComboset Combos class  is then used to generate all pairs of jugs from the available number.  If the first of the pair could pour an amount >0 into the 2nd of the pair, a graph edge is generated connecting the two.

Once the nodes and edges have been built, TComboboxes,  StartCB and EndCb,  are loaded with the available nodes from which the user will select starting and ending jar configurations.  When the start configuration is changed in StartCB, an attempt is made to evaluate the result for all goal nodes in EndCB.  For four jar cases with large capacities, this process might take a long time, so a "Stop" button is provided to abort the process. The results of the solution search are used to pre-calculate "Unsolvable" goals which are flagged in the end goal text and "best solution" move counts, which are retained as objects with each goal entry and displayed when the "Show best solutions" checkbox is checked.  The best goal count is also used to modify the message given when a solution is found manually  ("Congratulations!" or "Good, but it can be solved in fewer moves")

User play is controlled by an OnDrawCell exit in a TStringGrid which draws scaled images of the jugs (they look more like beakers, but oh well).   Users click on the images define from and to jugs and the move is displayed in a Tmemo display area.

Whew!  700 lines of code is enough to put the program in the Advanced category.

Addendum: February 18, 2010:  A 7 jug problem sent by a viewer led to a major rewrite of this program over the last month.  Version 3 solves cases much faster than the earlier version and handles problems up to 7 jugs in size.  Melanie's 7 jug problem is displayed as the default for that size and takes a couple of minutes to find the shortest solution which requires 48 moves!  Most other cases I've tried find the solution in under a second.   Internally, Dijkstra and the Graphlist are longer used.  That method required building the complete search state graph before running the search.  For large cases this is not feasible.  The search now is a breadth first search for the goal configuration but generating only those goal states which derive from the starting configuration.  There were some interesting problems to solve  in removing the recursive search procedure used in the prior version.  See the source code or write if you are interested in the details.

Addendum: April 21, 2010: A second type of Water Jug puzzle, which I am calling "Open" puzzles, was introduced today with Version 4.  The original puzzles implemented here are "Closed" in the sense that water may neither be added or subtracted from the initial amounts while solving.  "Open" puzzles allow water to be added or subtracted by filling or emptying any jug in addition to pouring from one to another.  Searching "Water jug puzzles" on the web will return a number of problems of this type, including a classic one from a "Die Hard" movie in which 4 gallons of water must be obtained when only jars with capacities or 3 and 5 gallons are available.