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Example grid for a cross-figure puzzle with some answers filled in. A cross-figure (also variously called cross number puzzle or figure logic) is a puzzle similar to a crossword in structure, but with entries that consist of numbers rather than words, where individual digits are entered in the blank cells.
Thus a single line can be drawn connecting all nine dots—which would appear as three lines in parallel on the paper, when flattened out. [18] It is also possible to fold the paper flat, or to cut the paper into pieces and rearrange it, in such a way that the nine dots lie on a single line in the plane (see fold-and-cut theorem). [17]
Dots and boxes is a pencil-and-paper game for two players (sometimes more). It was first published in the 19th century by French mathematician Édouard Lucas, who called it la pipopipette. [1] It has gone by many other names, [2] including dots and dashes, game of dots, [3] dot to dot grid, [4] boxes, [5] and pigs in a pen. [6]
Add the clues together, plus 1 for each "space" in between. For example, if the clue is 6 2 3, this step produces the sum 6 + 1 + 2 + 1 + 3 = 13. Subtract this number from the total available in the row (usually the width or height of the puzzle). For example, if the clue in step 1 is in a row 15 cells wide, the difference is 15 - 13 = 2.
With the bent hypotenuse, the first figure actually occupies a combined 32 units, while the second figure occupies 33, including the "missing" square. The amount of bending is approximately 1 / 28 unit (1.245364267°), which is difficult to see on the diagram of the puzzle, and was illustrated as a graphic. Note the grid point where the ...
An example of the scintillating grid illusion. Dark dots seem to appear and disappear at intersections. The scintillating grid illusion is an optical illusion, discovered by E. and B. Lingelbach and M. Schrauf in 1994. [2] It is often considered a variation of the Hermann grid illusion but possesses different properties. [2] [3]
Quincunx antialiasing samples scenes at the corners and centers of each pixel. These five sample points, in the shape of a quincunx, are combined to produce each displayed pixel. However, samples at the corner points are shared with adjacent pixels, so the number of samples needed is only twice the number of displayed pixels. [3]
This construction, new at the time, led him to many powerful results and methods. For example, he was able to show that all but ten Dehn surgeries on the figure-eight knot resulted in non-Haken, non-Seifert-fibered irreducible 3-manifolds; these were the first such examples. Many more have been discovered by generalizing Thurston's construction ...