What's the rarest 50p?

As a keen collector of British coins, I'm often intrigued by the rarity and value of certain denominations. Among the various coins in circulation, the 50-pence coin has always fascinated me due to its unique designs and potential collectability. Could you elaborate on what makes a particular 50p coin the rarest? Is it the limited mintage, the scarcity in circulation, or perhaps the unique design? Additionally, could you provide some examples of the rarest 50p coins in recent years and explain why they command such high premiums among collectors? The answer to this question would not only satisfy my curiosity but also guide other enthusiasts in their quest for rare coins.