The On-Line Encyclopedia of Bongard Problems

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Straight line parallel to line joining the ends of curve AND sliding the straight line suitably can make a closed area vs. straight line perpendicular to line joining the ends of curve.

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	CROSSREFS	`Adjacent-numbered pages: BP234 BP235 BP236 BP237 BP238 * BP240 BP241 BP242 BP243 BP244`
	KEYWORD	`traditional`
	CONCEPT	`line_or_curve_endpoint (info \| search),` `length_line_or_curve (info \| search),` `imagined_line_or_curve (info \| search),` `imagined_entity (info \| search),` `imagined_motion (info \| search),` `motion (info \| search),` `same_feature (info \| search),` `same (info \| search),` `parallel (info \| search),` `perpendicular (info \| search)`
	AUTHOR	`Harry E. Foundalis`

BP384

Square number of dots vs. non-square number of dots.

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	COMMENTS	`All examples in this Problem are a collection of dots.` `An equivalent solution is "Dots can be arranged into a square lattice whose convex hull is a square vs. not so". - Leo Crabbe, Aug 01 2020`
	CROSSREFS	`Adjacent-numbered pages: BP379 BP380 BP381 BP382 BP383 * BP385 BP386 BP387 BP388 BP389`
	EXAMPLE	`A single dot fits because 1 = 11.` `A pair of dots does not fit because there is no integer x such that 2 = xx.`
	KEYWORD	`nice, precise, allsorted, number, math, left-narrow, left-null, help, traditional, preciseworld, collection`
	CONCEPT	`square_number (info \| search)`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

BP209

No objects are centered at the center of the box vs. one object is centered at the center of the box.

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	CROSSREFS	`Adjacent-numbered pages: BP204 BP205 BP206 BP207 BP208 * BP210 BP211 BP212 BP213 BP214`
	KEYWORD	`noisy, boundingbox, traditional, absoluteposition`
	CONCEPT	`bounding_box (info \| search),` `existence (info \| search),` `number (info \| search),` `center_bounding_box (info \| search),` `center (info \| search)`
	AUTHOR	`Giuseppe Insana`

BP939

Optical illusions vs. not so.

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	COMMENTS	`Panels on the left hand side contain geometrical objects that appear distorted (to a human) due to surrounding information.`
	REFERENCE	`Vicente Sierra-Vázquez & Ignacio Serrano-Pedraza, Application of Riesz transforms to the isotropic AM-PM decomposition of geometrical-optical illusion images, April 2010, Figure 1.`
	CROSSREFS	`Adjacent-numbered pages: BP934 BP935 BP936 BP937 BP938 * BP940 BP941 BP942 BP943 BP944`
	KEYWORD	`fuzzy, anticomputer, subjective, contributepairs, invalid, experimental`
	CONCEPT	`length_line_or_curve (info \| search),` `visual_illusion (info \| search)`
	AUTHOR	`Leo Crabbe`

BP1261

Exactly one connected white region for each possible way shapes can overlap vs. not so.

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	COMMENTS	`All left-sorted examples have 2^n distinct white regions (counting the background), where n is the number of shapes in the image.`
	REFERENCE	`https://en.wikipedia.org/wiki/Venn_diagram`
	CROSSREFS	`Similar solution to BP1262, except there overlaps do not have to be unique.` `Adjacent-numbered pages: BP1256 BP1257 BP1258 BP1259 BP1260 * BP1262 BP1263 BP1264 BP1265 BP1266`
	KEYWORD	`notso, left-narrow, left-null, perfect`
	CONCEPT	`exists_one (info \| search)`
	AUTHOR	`Leo Crabbe`

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`

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"`

BP277

One object vs. two or more objects.

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	CROSSREFS	`Adjacent-numbered pages: BP272 BP273 BP274 BP275 BP276 * BP278 BP279 BP280 BP281 BP282`
	KEYWORD	`number`
	CONCEPT	`number (info \| search)`
	WORLD	`arial_font_symbols [smaller \| same \| bigger]`
	AUTHOR	`Merse E. Gáspár`

BP527

Each black filled circle belongs to exactly one large circle outline vs. not so.

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	COMMENTS	`On the left, circles define a partition of the dots.`
	CROSSREFS	`Adjacent-numbered pages: BP522 BP523 BP524 BP525 BP526 * BP528 BP529 BP530 BP531 BP532`
	KEYWORD	`precise, allsorted, traditional`
	CONCEPT	`circle (info \| search),` `exists_one (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1120

No same-sized copies of self overlap vs. distinct same-sized copies overlap.

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	COMMENTS	`With mathematical jargon:` `No distinct same-sized copies of self overlap on a subset with positive measure in the Hausdorff measure using the Hausdorff dimension.` `For a covering of a fractal by finitely many scaled down copies of itself, the condition of that no two have an intersection with positive measure is equivalent to the condition that the Hausdorff dimension coincides with the similarity dimension.` `(There is another similar condition in this context called the "open set condition" which implies this but is not equivalent. The open set condition is equivalent to the condition that the Hausdorff measure using the similarity dimension is nonzero.)`
	REFERENCE	`https://en.wikipedia.org/wiki/Hausdorff_dimension` `https://en.wikipedia.org/wiki/Open_set_condition`
	CROSSREFS	`Adjacent-numbered pages: BP1115 BP1116 BP1117 BP1118 BP1119 * BP1121 BP1122 BP1123 BP1124 BP1125`
	KEYWORD	`challenge, perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search),` `overlap (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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