Penny Napoleonic Wars prompted the use of base metals in 1797. Despite the decimalization of currencies in the United States and, later, throughout the British Commonwealth, the name remains in informal use. No penny is currently formally subdivided, although farthings (¼ d.), halfpennies, and half cents have previously been minted and the mill (1/10¢) remains in use as a unit of account in some contexts. "Penny" is first attested in a 1394 Scots text, a variant of Old English ', a development of numerous variations including ', ', and '. The etymology of the term "penny" is uncertain, although cognates are
Penny graph that don't touch each other. By the four-color theorem, , and the improved bound was proven by Swanepoel. In the other direction, Pach and Tóth proved that . As of 2013, these remained the best bounds known for this problem. Penny graph In geometric graph theory, a penny graph is a contact graph of unit circles. That is, it is an undirected graph whose vertices can be represented by unit circles, with no two of these circles crossing each other, and with two adjacent vertices if and only if they are represented by tangent circles. More simply, they are the