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"`

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`

BP1215

Axis of symmetry along the NW/SE diagonal vs. not.

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	CROSSREFS	`BP1206 is the same solution but with the vertical axis instead of the NW/SE diagonal.` `BP1207 is the same solution but with the horizontal axis instead of the NW/SE diagonal.` `Adjacent-numbered pages: BP1210 BP1211 BP1212 BP1213 BP1214 * BP1216 BP1217 BP1218 BP1219 BP1220`
	KEYWORD	`stub, notso, handed, leftright, updown, rotate, left-narrow, traditional`
	CONCEPT	`diagonal (info \| search),` `symmetry_axis (info \| search),` `symmetry (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP891

Dots can be connected to create one triangle within another vs. not so.

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	COMMENTS	`Or equivalently, "3 dots within boundary of convex hull vs. not so". - Leo Crabbe, Aug 01 2020`
	CROSSREFS	`Adjacent-numbered pages: BP886 BP887 BP888 BP889 BP890 * BP892 BP893 BP894 BP895 BP896`
	KEYWORD	`nice, precise, allsorted, preciseworld`
	CONCEPT	`convex_hull (info \| search),` `triangle (info \| search)`
	WORLD	`6_dots [smaller \| same \| bigger]`
	AUTHOR	`Cameron Fetter`

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`

BP548

Due to pixelation, it is ambiguous whether the middle small shape is a resized version of the left shape or the right shape vs. there is a clear choice.

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	COMMENTS	`A strange Problem relying on pixelation of digital images.`
	CROSSREFS	`Adjacent-numbered pages: BP543 BP544 BP545 BP546 BP547 * BP549 BP550 BP551 BP552 BP553`
	KEYWORD	`creativeexamples, pixelperfect, structure, orderedtriplet, experimental`
	CONCEPT	`imperfection_small (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP1110

The process that turns one object into the other is the same both ways vs. the process changes depending on which object is chosen as the starting point.

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	REFERENCE	`https://en.wikipedia.org/wiki/Duality_(mathematics)` `https://en.wikipedia.org/wiki/Involution_(mathematics)`
	CROSSREFS	`This is a special case of BP841 and a generalisation of BP822.` `Adjacent-numbered pages: BP1105 BP1106 BP1107 BP1108 BP1109 * BP1111 BP1112 BP1113 BP1114 BP1115`
	KEYWORD	`nice, abstract, math, anticomputer, creativeexamples, left-narrow, unorderedpair, rules, miniworlds, dithering`
	CONCEPT	`function (info \| search)`
	AUTHOR	`Leo Crabbe`

BP330

1-D curve vs. 2-D shape.

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	CROSSREFS	`Adjacent-numbered pages: BP325 BP326 BP327 BP328 BP329 * BP331 BP332 BP333 BP334 BP335`
	KEYWORD	`traditional`
	CONCEPT	`dimensionality (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

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