The On-Line Encyclopedia of Bongard Problems

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If "distance" is taken to be the sum of horizontal and vertical distances between points, the 3 points are equidistant from each other vs. not so.

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	COMMENTS	`In other words, we take the distance between points (a,b) and (c,d) to be equal to \|c-a\| + \|d-b\|, or, in other words, the distance of the shortest path between points that travels along grid lines. In mathematics, this way of measuring distance is called the 'taxicab' or 'Manhattan' metric. The points on the left hand side form equilateral triangles in this metric.` `⠀` `An alternate (albeit more convoluted) solution that someone may arrive at for this Problem is as follows: The triangles formed by the points on the left have some two points diagonal to each other (in the sense of bishops in chess), and considering the corresponding edge as their base, they also have an equal height. However, this was proven to be equivalent to the Manhattan distance answer by Sridhar Ramesh. Here is the proof:` `⠀` `An equilateral triangle amounts to points A, B, and C such that B and C lie on a circle of some radius centered at A, and the chord from B to C is as long as this radius.` A Manhattan circle of radius R is a turned square, ♢, where the Manhattan distance between any two points on opposite sides is 2R, and the Manhattan distance between any two points on adjacent sides is the larger distance from one of those points to the corner connecting those sides. Thus, to get two of these points to have Manhattan distance R, one of them must be a midpoint of one side of the ♢ (thus, bishop-diagonal from its center) and the other can then be any point on an adjacent side of the ♢ making an acute triangle with the aforementioned midpoint and center.
	CROSSREFS	`Adjacent-numbered pages: BP929 BP930 BP931 BP932 BP933 * BP935 BP936 BP937 BP938 BP939`
	KEYWORD	`hard, allsorted, solved, left-finite, right-finite, perfect, pixelperfect, unorderedtriplet, finishedexamples`
	CONCEPT	`triangle (info \| search)`
	WORLD	`3_dots_on_square_grid [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP935

Shapes have equal area vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP930 BP931 BP932 BP933 BP934 * BP936 BP937 BP938 BP939 BP940`
	KEYWORD	`nice, precise, allsorted, unstable, left-narrow, perfect, pixelperfect, unorderedpair`
	CONCEPT	`area (info \| search)`
	WORLD	`2_fill_shapes [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP966

Even number of white regions vs. odd number of white regions.

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	COMMENTS	`See BP889 for the version in which the background doesn't count.`
	CROSSREFS	`Adjacent-numbered pages: BP961 BP962 BP963 BP964 BP965 * BP967 BP968 BP969 BP970 BP971`
	KEYWORD	`nice, boundingbox, number, right-null, perfect, pixelperfect, traditional, bordercontent`
	CONCEPT	`separated_regions (info \| search),` `even_odd (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1008

The 26th from the left, 63rd from the top pixel is black versus white.

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	COMMENTS	`This is a typical kind of joke answer people give for Bongard Problems when they cannot find an answer.`
	CROSSREFS	`Adjacent-numbered pages: BP1003 BP1004 BP1005 BP1006 BP1007 * BP1009 BP1010 BP1011 BP1012 BP1013`
	KEYWORD	`less, dual, arbitrary, handed, leftright, updown, boundingbox, blackwhite, antihuman, right-null, perfect, pixelperfect, help, experimental, funny, absoluteposition, bordercontent`
	CONCEPT	`specificity (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1088

Top piece can fit into bottom piece without leaving any overhangs vs. not so

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	COMMENTS	`The piece does not need to slide in from above, unlike the game Tetris.`
	CROSSREFS	`Adjacent-numbered pages: BP1083 BP1084 BP1085 BP1086 BP1087 * BP1089 BP1090 BP1091 BP1092 BP1093`
	KEYWORD	`perfect, pixelperfect`
	AUTHOR	`William B Holland`

BP1089

Equal number of black and white pixels vs. not

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	CROSSREFS	`Adjacent-numbered pages: BP1084 BP1085 BP1086 BP1087 BP1088 * BP1090 BP1091 BP1092 BP1093 BP1094`
	KEYWORD	`nice, precise, allsorted, boundingbox, right-null, perfect, pixelperfect, help, preciseworld, bordercontent, blackwhiteinvariant`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

BP1090

Top piece can slide into bottom area in such that there is an unbroken black horizontal region vs. not so.

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	COMMENTS	`Successful moves in Tetris.`
	CROSSREFS	`Adjacent-numbered pages: BP1085 BP1086 BP1087 BP1088 BP1089 * BP1091 BP1092 BP1093 BP1094 BP1095`
	KEYWORD	`precise, teach, culture, pixelperfect`
	AUTHOR	`Leo Crabbe`

BP1093

"Inverted symmetry" present vs. not

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	CROSSREFS	`Adjacent-numbered pages: BP1088 BP1089 BP1090 BP1091 BP1092 * BP1094 BP1095 BP1096 BP1097 BP1098`
	KEYWORD	`nice, precise, allsorted, boundingbox, left-narrow, right-null, perfect, pixelperfect, preciseworld, bordercontent, blackwhiteinvariant`
	CONCEPT	`black_white_inversion (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

BP1104

Vertically centered versus horizontally centered

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	COMMENTS	`It is easier to notice that the left examples are vertically centered than that the right examples are horizontally centered. - Aaron David Fairbanks, Dec 27 2022`
	CROSSREFS	`Adjacent-numbered pages: BP1099 BP1100 BP1101 BP1102 BP1103 * BP1105 BP1106 BP1107 BP1108 BP1109`
	KEYWORD	`easy, nice, precise, minimal, boundingbox, perfect, pixelperfect, traditional, finishedexamples, preciseworld, absoluteposition, unstableworld`
	CONCEPT	`center_bounding_box (info \| search),` `center (info \| search)`
	WORLD	`horizontal_line_segment [smaller \| same \| bigger]`
	AUTHOR	`Widad Dabbas`

BP1131

One shape can be totally obscured by the other vs. neither shape can be obscured.

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	COMMENTS	`Rotation of shapes is not required for any left-hand panels, but it should not change any example's sorting if it is considered.`
	CROSSREFS	`Adjacent-numbered pages: BP1126 BP1127 BP1128 BP1129 BP1130 * BP1132 BP1133 BP1134 BP1135 BP1136`
	KEYWORD	`nice, precise, allsorted, pixelperfect, unorderedpair`
	CONCEPT	`overlap (info \| search)`
	WORLD	`2_shapes [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

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