The On-Line Encyclopedia of Bongard Problems

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BP285

The centers (barycenters) of the objects are collinear vs. the centers (barycenters) of the objects are not collinear.

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	CROSSREFS	`Adjacent-numbered pages: BP280 BP281 BP282 BP283 BP284 * BP286 BP287 BP288 BP289 BP290`
	KEYWORD	`traditional, finishedexamples`
	CONCEPT	`collinear (info \| search)`
	AUTHOR	`"Lewis"`

BP304

At least one (interior or exterior) right angle vs. no right angle (either interior or exterior).

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	CROSSREFS	`Adjacent-numbered pages: BP299 BP300 BP301 BP302 BP303 * BP305 BP306 BP307 BP308 BP309`
	KEYWORD	`traditional, finishedexamples`
	WORLD	`polygon_outline [smaller \| same \| bigger]`
	AUTHOR	`"Lewis"`

BP328

All sides are equal vs. all angles are equal.

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	CROSSREFS	`Adjacent-numbered pages: BP323 BP324 BP325 BP326 BP327 * BP329 BP330 BP331 BP332 BP333`
	KEYWORD	`nice, stretch, left-narrow, right-narrow, traditional, finishedexamples, both`
	CONCEPT	`angle (info \| search),` `length_line_or_curve (info \| search),` `same_feature (info \| search),` `same (info \| search)`
	WORLD	`polygon_outline [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP329

Regular polygon vs. not regular polygon.

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	CROSSREFS	`Adjacent-numbered pages: BP324 BP325 BP326 BP327 BP328 * BP330 BP331 BP332 BP333 BP334`
	KEYWORD	`precise, allsorted, stretch, traditional, finishedexamples, preciseworld`
	CONCEPT	`angle (info \| search),` `same_feature (info \| search),` `same (info \| search)`
	WORLD	`polygon_outline [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP342

Exactly one axis of symmetry vs. either zero or more than one axis of symmetry.

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	CROSSREFS	`Adjacent-numbered pages: BP337 BP338 BP339 BP340 BP341 * BP343 BP344 BP345 BP346 BP347`
	KEYWORD	`traditional, finishedexamples`
	CONCEPT	`exists_one (info \| search),` `imagined_line_or_curve (info \| search),` `imagined_entity (info \| search),` `symmetry_axis (info \| search),` `symmetry (info \| search)`
	WORLD	`shape_outline [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP801

Number pointed to on number line is "important" mathematical constant vs. not so.

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	COMMENTS	`This is the "harder version" of BP505.`
	CROSSREFS	`Adjacent-numbered pages: BP796 BP797 BP798 BP799 BP800 * BP802 BP803 BP804 BP805 BP806`
	KEYWORD	`hard, less, abstract, math, subjective, challenge, right-unknowable, collective, experimental, finishedexamples`
	AUTHOR	`Aaron David Fairbanks`

BP934

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`

BP1017

Line segments linking same-coloured dots would intersect vs. not so.

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	CROSSREFS	`This is a less noisy version of BP261.` `Adjacent-numbered pages: BP1012 BP1013 BP1014 BP1015 BP1016 * BP1018 BP1019 BP1020 BP1021 BP1022`
	KEYWORD	`easy, nice, precise, allsorted, perfect, traditional, finishedexamples, preciseworld`
	CONCEPT	`lines_coincide (info \| search),` `imagined_line_or_curve (info \| search),` `imagined_entity (info \| search),` `overlap (info \| search)`
	AUTHOR	`Leo Crabbe`

BP1056

Blank image vs. nothing.

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	COMMENTS	`Two kinds of "nothing".`
	CROSSREFS	`See also BP1219, "blank image vs. image of blank square".` `Adjacent-numbered pages: BP1051 BP1052 BP1053 BP1054 BP1055 * BP1057 BP1058 BP1059 BP1060 BP1061`
	KEYWORD	`left-finite, right-finite, left-full, right-full, left-null, finished, invalid, experimental, funny, finishedexamples`
	CONCEPT	`empty (info \| search),` `existence (info \| search),` `zero (info \| search)`
	WORLD	`blank_image [smaller \| same \| bigger] zoom in left (blank_image) \| zoom in right (nothing)`
	AUTHOR	`Aaron David Fairbanks`

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`

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