Gokigen Naname is played on a rectangular grid in which numbers in circles appear at some of the intersections on the grid.
The object is to draw diagonal lines in each cell of the grid, such that the number in each circle equals the number of lines extending from that circle. Additionally, it is forbidden for the diagonal lines to form an enclosed loop. Unlike many of Nikoli's similar puzzles, such as Hashiwokakero, a single network of lines is not required.
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Immediate deductions from single numbers.
(top) Either 0 or 1 in a corner specifies that cell.
(middle row) A number at an edge determines both adjacent cells unless it is 1, in which case it is known that the two share a common value which is not yet determined.
(bottom) A lone number away from the edge gives no information unless it is 4. (0 cannot occur away from an edge, because it would force a closed loop)
Deductions from two numbers. When immediately adjacent and not at the edge, either two 1s or two 3s require the intervening two cells to share the same value. When it is known that a 1 adjoins two cells with the same value, one must contact that position; therefore the opposite two cells must form a semicircle around the 1. When a 3 adjoins two cells with the same value, one does not contact it, so the opposite two cells must both meet that position. A cell away from the edge cannot connect two 1s at opposite corners, because that would force a closed loop.
As a direct consequence of the rules, all diagonal lines must be connected to the edges of the grid by other such lines. If not, one would immediately create a closed circuit around this line. Therefore, if a network of lines does not touch the edge of the grid, and only has one place where it can touch the edge, then it must do so.
- David J. Bodycombe (4 December 2007). The Riddles of the Sphinx: And the Puzzles, Word Games, Brainteasers, Conundrums, Quizzes, Mysteries, Codes and Ciphers That Have Baffled, Entertained and Confused the World Over the Last 100 Years. Penguin Group USA. ISBN 978-0-14-311275-4. Retrieved 14 August 2013.