The On-Line Encyclopedia of Bongard Problems

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When two players alternate coloring regions, either can force connection from top edge to bottom edge vs. either can force connection from left edge to right edge.

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

	REFERENCE	`https://en.wikipedia.org/wiki/Hex_(board_game)`
	CROSSREFS	`Adjacent-numbered pages: BP1240 BP1241 BP1242 BP1243 BP1244 * BP1246 BP1247 BP1248 BP1249 BP1250`
	KEYWORD	`hard, precise, convoluted, dual, rotate, boundingbox, hardsort, challenge, proofsrequired, bordercontent`
	AUTHOR	`Aaron David Fairbanks`

BP1241

Any point contained in (arbitrarily) smaller version of self vs. not so.

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	COMMENTS	`Note if any point is contained in some smaller version of the whole, then any point is contained in arbitrarily smaller versions of the whole.` `It isn't possible to unambiguously communicate in a picture whether or not a few specific points are included in the fractal. The pictures are interpreted as what is intuitively simplest. To make matters less ambiguous, all the fractals here contain all points arbitrarily close to points in them. (They are topologically closed. See also BP1239.)` `The left hand side of this is a stronger condition than the left hand side of BP1116.`
	CROSSREFS	`Adjacent-numbered pages: BP1236 BP1237 BP1238 BP1239 BP1240 * BP1242 BP1243 BP1244 BP1245 BP1246`
	KEYWORD	`notso, perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	WORLD	`connected_fractal [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP1240

Contains an un-nested white rectangle vs doesn't

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

	CROSSREFS	`Adjacent-numbered pages: BP1235 BP1236 BP1237 BP1238 BP1239 * BP1241 BP1242 BP1243 BP1244 BP1245`
	CONCEPT	`existence (info \| search)`
	AUTHOR	`Leo Crabbe`

BP1239

Fractal topologically closed (each white point has a neighborhood of pure white surrounding it) vs. not

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

	COMMENTS	`It isn't possible to unambiguously communicate whether or not a few specific points are missing.`
	CROSSREFS	`Adjacent-numbered pages: BP1234 BP1235 BP1236 BP1237 BP1238 * BP1240 BP1241 BP1242 BP1243 BP1244`
	KEYWORD	`notso, perfect, infinitedetail`
	CONCEPT	`fractal (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1238

Fractal with hole vs. fractal with no hole (simply connected).

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

	CROSSREFS	`Adjacent-numbered pages: BP1233 BP1234 BP1235 BP1236 BP1237 * BP1239 BP1240 BP1241 BP1242 BP1243`
	KEYWORD	`notso, perfect, infinitedetail`
	CONCEPT	`closed_open (info \| search),` `existence (info \| search),` `fractal (info \| search),` `hole (info \| search),` `loop (info \| search)`
	WORLD	`connected_fractal_self_tile [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP1237

Connected fractal vs. disconnected fractal.

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

	CROSSREFS	`Adjacent-numbered pages: BP1232 BP1233 BP1234 BP1235 BP1236 * BP1238 BP1239 BP1240 BP1241 BP1242`
	KEYWORD	`notso, perfect, infinitedetail`
	CONCEPT	`connected_component (info \| search),` `fractal (info \| search)`
	WORLD	`fractal_self_tile [smaller \| same \| bigger] zoom in left (connected_fractal_self_tile)`
	AUTHOR	`Jago Collins`

BP1233

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

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

	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`

BP1232

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

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

	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`

BP1230

Invariant under scaling by a certain amount vs. not.

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

	COMMENTS	`Each example on the left is imagined to continue beyond the bounding box.`
	CROSSREFS	`Adjacent-numbered pages: BP1225 BP1226 BP1227 BP1228 BP1229 * BP1231 BP1232 BP1233 BP1234 BP1235`
	KEYWORD	`stub, notso, perfect, infinitedetail, left-couldbe, bordercontent, blackwhiteinvariant`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `symmetry (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1229

Translational symmetry vs. no translational symmetry.

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

	COMMENTS	`Each example on the left is imagined to continue beyond the bounding box.`
	CROSSREFS	`Adjacent-numbered pages: BP1224 BP1225 BP1226 BP1227 BP1228 * BP1230 BP1231 BP1232 BP1233 BP1234`
	KEYWORD	`stub, notso, seemslike`
	CONCEPT	`symmetry (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

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