These directions are for Windows. 
There are two methods to run the game: with or without a command prompt.

First, install the JRE - you can just use the install defaults.
Then, unzip the game WITH DIRECTORIES (if using WinZip, make sure "Use Folder Names" is checked) to any directory.

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With a command prompt:
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Open a prompt and change to the directory of the DMJava game (wherever you unzipped it - there should be a bunch of .class files there and some pics).

Add the JRE bin directory to your path by typing:
set PATH="%PATH%;C:\Program Files\Java\j2re1.4.0\bin"
(your path to the bin directory may differ)

Run the game by typing:
java dmnew

Run the editor by typing:
java DMEditor

Note that this is CASE SENSITIVE! "dmnew" is all lower case, while "DMEditor" has the first three letters capitalized.
(You can set up batch files or modify your autoexec.bat to avoid all this typing every time.)

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Without a command prompt:
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Create a shortcut in the Start Menu, on the desktop, or wherever you want.

One way is to right click, go to New ->, and click Shortcut.
It will ask for a command line. Browse for or type the path to javaw.exe. (It will be in the bin directory of the JRE.) After it, add "-cp. dmnew" or "-cp . DMEditor", depending on which you are creating a shortcut for. Depending on where you installed things, the shortcut commmand line may look like
"C:\Program Files\Java\j2re1.4.0\bin\javaw.exe" "-cp . dmnew"

Click Next.

Give the shortcut a name ("Dungeon Master Java" or something).

Click Finish.
(You aren't really finished. If you double click it now, it will say it can't find the main class file.)

Right click the newly created shortcut and select "Properties". Change the "Start In" field to be the DM Java game directory (wherever you unzipped it). For example, "D:\DMJava"

Now you are done. Double click to run.