The On-Line Encyclopedia of Bongard Problems

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BP319

The number of dots in one cluster is a multiple of the other vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP314 BP315 BP316 BP317 BP318 * BP320 BP321 BP322 BP323 BP324`
	KEYWORD	`math, unorderedpair, traditional`
	CONCEPT	`multiple_of (info \| search),` `number (info \| search),` `dot (info \| search),` `proportional (info \| search)`
	WORLD	`two_dot_clusters [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP543

Image depicting infinitely many objects vs. image depicting finitely many objects.

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	COMMENTS	`Another solution is "evokes infinity vs. not so." Or "seen as having arbitrarily fine detail." All left examples show objects getting smaller and smaller and closer and closer together approaching some limit within the box.`
	CROSSREFS	`Adjacent-numbered pages: BP538 BP539 BP540 BP541 BP542 * BP544 BP545 BP546 BP547 BP548`
	KEYWORD	`nice, abstract, concept, infinitedetail`
	CONCEPT	`finite_infinite (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP163

Line connecting small shapes does not intersect large one vs. line connecting small shapes intersects large one.

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	CROSSREFS	`See BP1015 for a re-skinned version with dots instead of shapes.` `Adjacent-numbered pages: BP158 BP159 BP160 BP161 BP162 * BP164 BP165 BP166 BP167 BP168`
	KEYWORD	`traditional`
	CONCEPT	`imagined_line_or_curve (info \| search),` `imagined_entity (info \| search),` `object_line_overlap (info \| search),` `overlap (info \| search)`
	AUTHOR	`Harry E. Foundalis`

BP274

Notch from above (holds poured water) vs. no notch from above (does not hold poured water).

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	CROSSREFS	`Adjacent-numbered pages: BP269 BP270 BP271 BP272 BP273 * BP275 BP276 BP277 BP278 BP279`
	KEYWORD	`notso, physics`
	CONCEPT	`concave_convex_region (info \| search),` `downward_upward (info \| search),` `line_absolute_direction (info \| search),` `absolute_direction (info \| search),` `direction (info \| search),` `liquid_flow (info \| search),` `gravity (info \| search)`
	WORLD	`arial_font_centered_connected_symbol [smaller \| same \| bigger]`
	AUTHOR	`Merse E. Gáspár`

BP1223

Center square is black vs. center square is white.

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	COMMENTS	`All examples are a three by three grid of black or white squares.`
	CROSSREFS	`Adjacent-numbered pages: BP1218 BP1219 BP1220 BP1221 BP1222 * BP1224 BP1225 BP1226 BP1227 BP1228`
	KEYWORD	`stub, left-finite, right-finite, traditional, finishedexamples, fixedgrid, preciseworld`
	CONCEPT	`outlined_filled (info \| search),` `center (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP288

The sum of the ratios of the filled areas is 1 vs. the sum of the ratios of the filled areas is other than 1.

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

	CROSSREFS	`Adjacent-numbered pages: BP283 BP284 BP285 BP286 BP287 * BP289 BP290 BP291 BP292 BP293`
	KEYWORD	`nice, notso, traditional`
	CONCEPT	`addition (info \| search),` `outlined_filled (info \| search),` `ratio (info \| search),` `texture (info \| search)`
	AUTHOR	`"Lewis"`

BP45

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

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	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.

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	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`

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