The On-Line Encyclopedia of Bongard Problems

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The completed version of the collection is finite vs. the completed version of the collection is infinite.

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	COMMENTS	`Related to BP379 and BP792.` `An example with clusters of perfect numbers of dots would be sorted ambiguously, for the time being. Unfortunately the 3rd smallest perfect number is 496. - Leo Crabbe, Oct 18 2024`
	CROSSREFS	`See also BP870.` `Adjacent-numbered pages: BP375 BP376 BP377 BP378 BP379 * BP381 BP382 BP383 BP384 BP385`
	KEYWORD	`math, creativeexamples, traditional, rules, miniworlds, collection`
	CONCEPT	`finite_infinite (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

Any straight path will hit a black region vs. not so.

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	REFERENCE	`https://en.wikipedia.org/wiki/Opaque_set`
	CROSSREFS	`Adjacent-numbered pages: BP1244 BP1245 BP1246 BP1247 BP1248 * BP1250 BP1251 BP1252 BP1253 BP1254`
	KEYWORD	`notso, blackwhite, right-null, perfect, bordercontent`
	CONCEPT	`imagined_line_or_curve (info \| search)`
	AUTHOR	`Leo Crabbe`

Mouth concave upwards vs. Mouth convex upwards.

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	CROSSREFS	`Adjacent-numbered pages: BP210 BP211 BP212 BP213 BP214 * BP216 BP217 BP218 BP219 BP220`
	CONCEPT	`concave_convex_region (info \| search),` `downward_upward (info \| search),` `line_absolute_direction (info \| search),` `absolute_direction (info \| search),` `direction (info \| search)`
	AUTHOR	`Giuseppe Insana`

The numbers of dots in the two clusters are approximately equal (log scale) vs. the numbers of dots in the two clusters differ a lot.

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	COMMENTS	`The point is that, on the left, the two groups look about the same to the eye at first glance.` `In left examples the numbers are off by a factor of 7/6 or less vs. in right examples the numbers are off by a factor of 2 or greater.`
	CROSSREFS	`Adjacent-numbered pages: BP315 BP316 BP317 BP318 BP319 * BP321 BP322 BP323 BP324 BP325`
	KEYWORD	`nice, traditional`
	CONCEPT	`numerosity (info \| search),` `similar_number (info \| search),` `similar (info \| search)`
	WORLD	`two_dot_clusters [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

Acute angle vs. no acute angle.

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	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 224.`
	CROSSREFS	`Adjacent-numbered pages: BP28 BP29 BP30 BP31 BP32 * BP34 BP35 BP36 BP37 BP38`
	KEYWORD	`nice, stretch, finished, traditional, bongard`
	CONCEPT	`acute (info \| search),` `angle (info \| search),` `existence (info \| search)`
	WORLD	`Multiple options: mediumsize_centered_outline [smaller \| same \| bigger], shape_outline [smaller \| same \| bigger]`
	AUTHOR	`Mikhail M. Bongard`

Fractal tiles itself with uniformly scaled-down copies of itself vs. fractal tiles itself with stretched copies of itself.

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	COMMENTS	`"Self-similar" vs. "self-affine."`
	CROSSREFS	`Adjacent-numbered pages: BP525 BP526 BP527 BP528 BP529 * BP531 BP532 BP533 BP534 BP535`
	KEYWORD	`perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `self-reference (info \| search),` `tiling (info \| search)`
	WORLD	`fractal_self_tile_affine_allowed [smaller \| same \| bigger] zoom in left (fractal_self_tile)`
	AUTHOR	`Aaron David Fairbanks`

Lines plus triangles equal stars vs. lines minus triangles equal stars.

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	CROSSREFS	`Adjacent-numbered pages: BP228 BP229 BP230 BP231 BP232 * BP234 BP235 BP236 BP237 BP238`
	KEYWORD	`traditional`
	CONCEPT	`addition (info \| search),` `number (info \| search)`
	AUTHOR	`Peter Shanahan`

Bottom object in front of top objects in 3-D vs. bottom object behind top objects in 3-D.

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	CROSSREFS	`Adjacent-numbered pages: BP190 BP191 BP192 BP193 BP194 * BP196 BP197 BP198 BP199 BP200`
	KEYWORD	`experimental, dithering`
	CONCEPT	`3d_front_back (info \| search),` `3d_depth (info \| search),` `same_shape (info \| search),` `same (info \| search)`
	AUTHOR	`Harry E. Foundalis`

Can fold into tetragonal disphenoid ("isosceles tetrahedron") vs. cannot.

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	COMMENTS	`Which two sides are the long sides and which side is the short side, or equivalently which angles are the wider angles and which angle is the narrower angle, is the only relevant information to consider for each triangle. Triangles are all assumed isosceles and congruent to one another.` `All examples in this Problem feature four of these triangles connected by corners and/or edges.`
	CROSSREFS	`BP897 was conceived as a false solution for this.` `Adjacent-numbered pages: BP893 BP894 BP895 BP896 BP897 * BP899 BP900 BP901 BP902 BP903`
	KEYWORD	`hard, precise, allsorted, notso, math, preciseworld`
	CONCEPT	`triangle (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Molly C Klenzak`

Curves with at least one crossing vs. curves do not cross.

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	COMMENTS	`Based on BP30, "curve with exactly one self-crossing vs. curve without a self-crossing". This is just a slightly more general version.`
	CROSSREFS	`Adjacent-numbered pages: BP1227 BP1228 BP1229 BP1230 BP1231 * BP1233 BP1234 BP1235 BP1236 BP1237`
	EXAMPLE	`EX6881`
	KEYWORD	`stub, notso, traditional`
	CONCEPT	`x-crossing (info \| search),` `existence (info \| search)`
	WORLD	`crosscurves [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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