Under the dry stones of Hadrians Wall in northern England, excavators have found dice used by Roman Legionaires to gamble away their pay. Some of these dice have been found to be rigged. By filing off the corners of these bone cubes ever so slightly, it was possible to make certain combinations of numbers come up more often than others. This deception had to be a fairly tricky affair, since the dice don't know who is throwing them. Perhaps each tampered pair was matched with an honest pair which could be exchanged by the trickster at will. The altered dice were known as cater-trey dice.
The term catercorner to describe the altered state of the corners of certain dice derives from the medieval French term quattuor or four cornered, and may have had a vulgate Latin original, although the names given to the sides of dice have generally been thought to come from Old French originally. In Middle English, the French quatre became catre or catter when spoken by square jawed Saxons. Over time, the term cater-corner came to mean diagonal, and it came to be used geographically. Thus the baker's shop might be described as cater-corner to the butcher's shop in the market square, ie diagonally across from.
It is likely that the term came to America with the Elizabethans and was then altered as people from non-English homelands attempted to wrap their mouths around this strange configuration of letters. Thus, in America, we ended up with kitty-corner and catacorner. South of the Mason-Dixon Line, according to H. L. Menchen, the term catawampus was generally taken up for the same meaning though no one has yet dared to speculate on how that large a jump in base words was made. Maybe the original sounded too French to good ol' boys, who just found a more comfortable way to say it.
Somewhere along the line, the term has picked up a related meaning: out-of-kilter or discombobulated. This is a bit strange when you consider it. It is clear that dice that are unreliable might be seen as askew by mystified losers, but in fact a whittled pair is much more reliable than a square set, at least to the sharper using them skillfully with a matched honest set.
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