Suppose Tom is a knight. Then, Gladys is a knight and Tim is a liar. Since Gladys is a knight, Kate and Tom are both liars. However, Tom is a knight, so that is a contradiction. Therefore, Tom is a liar.

Suppose Kate is a liar. Then, Kate and Tom are both liars. That is what Gladys said, so she is a knight. Also, Kate is lying, so Tim is a liar. Then, Tom's statement is true, so he is a knight. However, we know that Tom is a liar, so Kate is a knight.

Since Kate is a knight, her statement is true, so Tim is also a knight. Also, Kate and Tom are not both liars, so Gladys is a liar.

Gladys:liar
Kate:knight
Tom:liar
Tim:knight

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