The conundrum is a 9 letter word, presented to contestants in the form of an anagram. The first person to buzz in and provide the answer scores 10 points. If neither contestant gets the answer within 30 seconds then it was often thrown over to the studio audience. A correct answer from an audience member won them a Countdown mug.
A crucial conundrum occurs when the two players are within ten points of each other before the final conundrum round, and hence the winner will be whichever player can solve it. Since 2003, the studio lights are dimmed on a crucial conundrum.
In the early days of the show, if the scores were level on a conundrum, the game was declared a draw and the contestants returned the next day for a rematch, except in the series finals, where tie-break conundrums were necessary. In the modern format, draws are resolved by a second "sudden death" conundrum. If the second conundrum goes unsolved, and the scores remain tied, a third conundrum is used (and so on, presumably, ad infinitum.) Only a few games have ever required a third conundrum, and fourth Conundrums are even rarer, with none ever having been televised.
If both Contestants offer an incorrect answer to a Conundrum in the 30 seconds, then the Countdown is stopped, and neither contestant gets another opportunity to buzz in.