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: Pure Loop

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?

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3 thoughts on “Puzzle 86: Scarcity

  1. Pingback: Sparsely Clued Puzzles | Puzzled by Titles

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