Arguably the first ever Rugby game was in Rugby school Worcestershire England, when William Webb Ellis picked up the football and ran with it. But the first organised game is between the houses of Rugby school (Houses being their boarding houses) after Webb Ellis and two others wrote a book documenting the rules of Rugby.
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Whilst the story of Web Ellis collecting the ball and running forward with it in a game of football occurred in 1823 the rules and therefore the Rugby Football Union did not start until much later however, the very first formalised rugby match where the "rules and laws" were laid down occurred in 1863 - The first recorded club game - Richmond Vs Blackheath - played in London ...
Rugby was invented in an all boys school in England so the first game of Rugby was probably played there