The crossword solution for the term 'LIKE THE CHESHIRE CAT' is 'SMILING'. The Cheshire Cat is a character from Lewis Carroll's novel Alice's Adventures in Wonderland. It is known for its wide, mischievous grin and its ability to disappear and reappear at will. Therefore, the phrase 'LIKE THE CHESHIRE CAT' can be interpreted as having a wide, mischievous smile, which is represented by the crossword solution 'SMILING'.