# Puzzle 86: Scarcity

Pure Loop Loop: Draw a loop that passes all white cells and no black cell such that the loop goes horizontally or vertically at all times and never touches or crosses itself.

Expected difficulty EasyAnswerComment/E-mail if you want a solution to be published

Puzzle 86: Scarcity
Pure Loop
(click to enlarge)

Page 12 of Chessics #12 states that a 8×8 Pure Loop puzzle only needs four black squares to assure uniqueness of the solution, and page 143 of The Games and Puzzles Journal #8-9 states that with an addition of rows and columns (and black squares), the solution can be extended. However, I don’t find anything about how many extra black squares are necessary, so here I’ll give an upper bound: a $4n \times 4n$ Pure Loop needs at most $2n$ black squares to ensure a unique solution. In fact, the result can be generalized to a rectangle: a $4n \times a$ Pure Loop needs at most $2n$ black squares to ensure a unique solution (although obviously this is weak if $a$ is small compared to $4n$). Is there anything stronger?