# A crossword has 20 clues, of which 12 have unique answers. How many solutions does the crossword have?

Aug 23, 2016

According to how the question is understood, it seems to me that the answer could be anything from $0$ upwards. With the most straightforward interpretation it would be ${2}^{8} = 256$

#### Explanation:

Here are some basic suppositions:

• A certain crossword puzzle has space for $20$ words of length greater than one letter, some horizontal, some vertical, in (roughly) square cells.
• There may be several cells which are black - taking no letter.
• There are cells where words intersect, resulting in some dependency between possible words.
• Associated with each of the $20$ words is a clue which taken on its own may be sufficient to determine a unique word (e.g. "The capital of France (5)") or may be satisfied by several possibilities (e.g. "A boy's name (6)").

Some further assumptions from the question would be:

• $12$ of the words have unique answers.
• $8$ of the words have two possible answers each.
• There is no interaction between these choices - i.e. the different possible answers only vary in cells that do not intersect with any other words. For example, the word with clue "French impressionist painter (5)" might have possible answers "Manet" or "Monet". If the second letter intersects with no other word then either would be a possibility in the completed crossword.

These conditions could be understood from the question in one of two different ways:

• The clues taken on their own are sufficient to determine $12$ words uniquely and the other $8$ up to $2$ possibilities each.
• The clues and intersections of the letters are sufficient to determine $12$ of the words, but leave $2$ possibilities for each of the remaining $8$ words.

With one of these assumptions, there are ${2}^{8} = 256$ possible solutions for the whole crossword puzzle.

These assumptions can break in several different ways.