[Verse 1] Ten pigeons fly to nine small holes At least one hole must share the load This truth seems simple, almost trite But watch how it brings math to light The power hides in choosing right What counts as pigeons in our sight [Chorus] More objects than the categories Abundance forces coincidence Define your pigeons, count the holes The overlap is evidence Pigeonhole principle shows When something's bound to coexist [Verse 2] Five points inside a unit square Divide it into quarters fair Four boxes for five points to fall At least one box must hold them all Two points now share a smaller space Within root two over two they're placed [Chorus] More objects than the categories Abundance forces coincidence Define your pigeons, count the holes The overlap is evidence Pigeonhole principle shows When something's bound to coexist [Bridge] The art lies not in proving why But seeing where the principle applies Creative framing is the key Transform the problem cleverly What are your pigeons, what are holes? The setup holds the magic roles [Verse 3] Take numbers in a sequence long N squared plus one elements strong Some rising up, some falling down A pattern waits to be found N plus one length monotone The pigeonhole seeds are sown [Chorus] More objects than the categories Abundance forces coincidence Define your pigeons, count the holes The overlap is evidence Pigeonhole principle shows When something's bound to coexist [Outro] From Ramsey theory's distant heights To Erdős' combinatorial sights The embryo of counting grows When abundance overflows Simple truth with complex reach That's what pigeonholes can teach
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