Learning pathOriginal reference
Lesson 7 of 8

Reject impossible prefixes early

Use local validity rules for balanced parentheses and nonattacking queens.

Understand the idea

Balanced parentheses maintain the numbers opened and closed. Add an opening bracket while opens < n; add a closing bracket only while closes < opens. For queens, place one queen per row while tracking occupied columns and the two diagonal families.

Trace a small example

With n = 2 pairs, the valid strings are (()) and ()(). A prefix beginning with a closing bracket cannot be repaired later. On a board, queens at (row, col) share a diagonal when row − col or row + col matches; offset row − col for nonnegative table indices.

Conditions and cost

Undo each occupied marker after returning from a queen choice. The pruning rule prevents invalid prefixes, but the search still branches. A single valid arrangement proves existence, not the number of arrangements.

Pause & predict

When may a parentheses generator append a closing bracket?

Choose an answer
Try it yourself

Turn the idea into code

How many balanced strings use three pairs? List them.

Give me a hint

Try each legal first continuation and keep the prefix rule.

Show the explained solution

There are 5: ((())), (()()), (())(), ()(()), and ()()(). Each has equal final counts and no prefix with more closes than opens.

Your notes

Ready for the next idea?

Mark complete when you can explain it yourself.

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