The On-Line Encyclopedia of Bongard Problems

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BP45

Outline figure on top of solid black figure vs. black figure on top of outline figure.

(edit; present; nest [left/right]; search; history)

	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 228.`
	CROSSREFS	`Adjacent-numbered pages: BP40 BP41 BP42 BP43 BP44 * BP46 BP47 BP48 BP49 BP50`
	KEYWORD	`dual, finished, traditional, viceversa, bongard`
	CONCEPT	`3d_front_back (info \| search),` `outlined_filled (info \| search),` `objects_overlap (info \| search),` `overlap (info \| search),` `texture (info \| search)`
	AUTHOR	`Mikhail M. Bongard`

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`

BP1233

Different curves cross each other vs. curves only cross themselves (or do not cross at all).

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	CROSSREFS	`Adjacent-numbered pages: BP1228 BP1229 BP1230 BP1231 BP1232 * BP1234 BP1235 BP1236 BP1237 BP1238`
	KEYWORD	`notso, traditional`
	CONCEPT	`distinguishing_crossing_curves (info \| search),` `x-crossing (info \| search)`
	WORLD	`smooth_crosscurves [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP952

Images of Bongard Problems about images of Bongard Problems about images of Bongard Problems vs. images of Bongard Problems not including images of Bongard Problems including images of Bongard Problems.

(edit; present; nest [left/right]; search; history)

	CROSSREFS	`See BP547 for the version with links to pages on the OEBP instead of images of Bongard Problems (miniproblems).` `Any left example in this BP will be a left example for BP1084.` `Adjacent-numbered pages: BP947 BP948 BP949 BP950 BP951 * BP953 BP954 BP955 BP956 BP957`
	KEYWORD	`meta (see left/right), miniproblems, funny, presentationinvariant`
	CONCEPT	`recursion (info \| search)`
	WORLD	`[smaller \| same \| bigger] zoom in left`
	AUTHOR	`Leo Crabbe`

BP368

There is a point that can see (in straight lines) all points vs. there is no point that can see (in straight lines) all points.

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	COMMENTS	`Left examples are called "star domains."`
	CROSSREFS	`See BP388 for whether a distinguished point can see all points.` `Adjacent-numbered pages: BP363 BP364 BP365 BP366 BP367 * BP369 BP370 BP371 BP372 BP373`
	KEYWORD	`nice, precise, unstable, perfect, traditional`
	CONCEPT	`all (info \| search),` `existence (info \| search),` `imagined_point (info \| search),` `imagined_line_or_curve (info \| search),` `imagined_entity (info \| search)`
	WORLD	`shape [smaller \| same \| bigger] zoom in left (fill_shape_with_seeing_point)`
	AUTHOR	`Aaron David Fairbanks`

BP128

Same objects inside and outside the large shape vs. not same objects inside and outside the large shape.

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	CROSSREFS	`Adjacent-numbered pages: BP123 BP124 BP125 BP126 BP127 * BP129 BP130 BP131 BP132 BP133`
	KEYWORD	`noisy, traditional`
	CONCEPT	`inside (info \| search),` `interior_exterior (info \| search),` `outlined_filled (info \| search),` `shape_cluster (info \| search),` `cluster (info \| search),` `same_number (info \| search),` `same (info \| search),` `texture (info \| search)`
	AUTHOR	`Douglas R. Hofstadter`

BP198

Stays in vs. escapes.

(edit; present; nest [left/right]; search; history)

	CROSSREFS	`Adjacent-numbered pages: BP193 BP194 BP195 BP196 BP197 * BP199 BP200 BP201 BP202 BP203`
	KEYWORD	`nice`
	CONCEPT	`line_or_curve_endpoint (info \| search),` `inside (info \| search),` `tracing_line_or_curve (info \| search)`
	AUTHOR	`Harry E. Foundalis`

BP106

Negative slope vs. positive slope.

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	COMMENTS	`All examples feature a white circle and a black circle. The answer refers to the slope of the line connecting the two circles.` `The coloring of the circles is irrelevant.`
	CROSSREFS	`Adjacent-numbered pages: BP101 BP102 BP103 BP104 BP105 * BP107 BP108 BP109 BP110 BP111`
	KEYWORD	`nice, noisy, dual, handed, rotate, orderedpair, traditional`
	CONCEPT	`positive_negative_slope (info \| search),` `line_slope (info \| search),` `imagined_line_or_curve (info \| search),` `imagined_entity (info \| search)`
	AUTHOR	`Douglas R. Hofstadter`

BP44

Small circles on different arcs vs. small circles on one arc.

(edit; present; nest [left/right]; search; history)

	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 228.`
	CROSSREFS	`Adjacent-numbered pages: BP39 BP40 BP41 BP42 BP43 * BP45 BP46 BP47 BP48 BP49`
	KEYWORD	`finished, traditional, bongard`
	CONCEPT	`between (info \| search),` `cusp (info \| search),` `on_line_or_curve (info \| search),` `separation_of_joined_objects (info \| search)`
	AUTHOR	`Mikhail M. Bongard`

BP79

A dark circle is closer to the outline circle than to the triangle vs. a dark circle is closer to the triangle than to the outline circle.

(edit; present; nest [left/right]; search; history)

	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 240.`
	CROSSREFS	`Adjacent-numbered pages: BP74 BP75 BP76 BP77 BP78 * BP80 BP81 BP82 BP83 BP84`
	KEYWORD	`spectrum, finished, traditional, continuous, viceversa, bongard`
	CONCEPT	`length_line_or_curve (info \| search),` `near_far (info \| search),` `outlined_filled (info \| search),` `texture (info \| search),` `triangle (info \| search)`
	AUTHOR	`Mikhail M. Bongard`

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