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Getting the bleeding edge
If you don't want to learn Git or the command line, the easiest way is to use GitHub's Git client, and just clone Mathics3/mathics-core (there is a button at the top of https://github.com/Mathics3/mathics-core that says "Clone in Mac" or "Clone in Windows" depending on your platform). You can then run or install Mathics Core from the directory it downloads.
On Linux, you will need to use the command line.
This guide is specifically for Linux and Mac OS X. On Windows, you can do the same thing using msysgit.
Basically, if you have never used Git before, you need to just install it either from your operating system's package manager (for Mac OS X, fink is recommended), or from http://git-scm.com/ and run
git clone https://github.com/Mathics3/mathics-core.git
in whatever directory you want to put it.
In addition to the Mathics source code, you'll also require two other libraries: python-dev and libsqlite3-dev.
For Ubuntu the command to install them is
apt-get install python-dev libsqlite3-dev python-setuptools
In the directory containing the Mathics source code run
cd mathics-core
python mathics/main.py
or you can install it using
make install
Assuming everything worked Mathics should be installed for your system. You can now launch the command line program
mathics
or start the server with
mathicsserver
and navigate to http://localhost:8000 with your favourite web browser.