A group of friends - Alex, Beth, Chris, and Dana - decide to play a game where they each write down a secret on a piece of paper and place it in a hat. They then draw a random secret from the hat and try to guess whose secret it is. To make the game more challenging, they agree not to reveal their secrets until everyone has guessed correctly. As they start playing, Alex draws Beth's secret, Beth draws Chris's secret, Chris draws Dana's secret, and Dana draws Alex's secret. Without revealing any information, how can they figure out whose secret belongs to whom? Provide a brief response.
Each person knows whose secret they have drawn and also knows their own secret. They can start by giving subtle hints about the secret they have drawn without revealing it fully. For each hint, they also listen to others' hints to see if any could apply to their own secret.

For example, Alex, who has drawn Beth's secret, can start by giving a hint that could apply to Beth's secret, but is vague enough not to give it away completely. Beth listens to the hint and considers if it sounds like it could be her secret, while also giving her own hint about Chris's secret.

As they each give and listen to hints, the group can start to make educated guesses about the ownership of each secret. If someone guesses incorrectly, they can use the process of elimination and the information gained from the hints to make a better-informed guess on the next round of hints.

The key is for everyone to pay attention to the hints and use their deductive reasoning, noting how others react to the hints given, to figure out which secret belongs to whom. This process continues with each person revising their guesses based on the new information until all have correctly guessed the secret they hold. Since they are not revealing secrets until everyone has guessed correctly, the game relies heavily on their ability to pick up on the subtleties in their friends' hints and reactions.