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Why "Free State"?

Coat of arms of the Free State of Bavaria

The term "Free State" (in German "Freistaat") comes from the 19th century as a literary translation of the word "republic", as opposed to "monarchy". As a matter of fact, it was used by Länder who had switched from a former hereditary monarchy to a people-elected government. After the German Revolution of November 1918, when Imperial Germany became the Weimar Republic, most of the German states within the German Reich called themselves a Free State. 

Nowadays, 3 of the 16 German Länder use the expression “Freistaat”: Bavaria (Freistaat Bayern), Saxony (Freistaat Saxen), and Thuringia (Freistaat Thüringen).

The Bavarian coat of arms unites the colours of Upper-Palatinate (top left), Franconia (top right), Swabia (bottom right) and Upper and Lower Bavaria (bottom left). The white-and-blue lozenges on the central shield are the colours of the Wittelsbach family, and are today's symbol of Bavaria itself, along with the golden lions.

Source & more: bayern.de/Wappen (in German),
Historical Dictionary of Bavaria (in German)