The On-Line Encyclopedia of Bongard Problems

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No straight path passes from a white region to a grey one, then to white again (without hitting black) vs. such a path exists.

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	CROSSREFS	`Any small amount of convexity along the grey border will cause an example to fit right.` `Adjacent-numbered pages: BP1243 BP1244 BP1245 BP1246 BP1247 * BP1249 BP1250 BP1251 BP1252 BP1253`
	KEYWORD	`precise, allsorted, unwordable, notso, perfect`
	CONCEPT	`imagined_line_or_curve (info \| search)`
	WORLD	`[smaller \| same \| bigger] zoom in left`
	AUTHOR	`Leo Crabbe`

BP1252

No entirely white paths from one side to another vs. one can find such a path.

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	COMMENTS	`The left-sorted patterns connect all four corners with a single black region. - Rhodri Prysor, Oct 16 2024`
	CROSSREFS	`Adjacent-numbered pages: BP1247 BP1248 BP1249 BP1250 BP1251 * BP1253 BP1254 BP1255 BP1256 BP1257`
	KEYWORD	`precise, allsorted, notso, blackwhite, right-null, bordercontent`
	AUTHOR	`Leo Crabbe`

BP1255

Shades of green vs. shades of red.

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	CROSSREFS	`Adjacent-numbered pages: BP1250 BP1251 BP1252 BP1253 BP1254 * BP1256 BP1257 BP1258 BP1259 BP1260`
	KEYWORD	`easy, precise, allsorted, minimal, arbitrary, color, continuous`
	AUTHOR	`Leo Crabbe`

BP1264

Any shape's axis of reflection is shared by another shape vs. not so.

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

	CROSSREFS	`Adjacent-numbered pages: BP1259 BP1260 BP1261 BP1262 BP1263 * BP1265 BP1266 BP1267 BP1268 BP1269`
	KEYWORD	`precise, allsorted, notso, left-narrow`
	CONCEPT	`symmetry (info \| search)`
	AUTHOR	`Leo Crabbe`

BP1266

Angles either only increase or only decrease as one moves along the curve vs. not so.

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	COMMENTS	`It does not matter which end of the curve one starts from, nor which "side" angles are measured from, so long as the side is kept consistent.`
	CROSSREFS	`Adjacent-numbered pages: BP1261 BP1262 BP1263 BP1264 BP1265 * BP1267 BP1268 BP1269 BP1270 BP1271`
	KEYWORD	`allsorted, notso, traditional`
	CONCEPT	`angle (info \| search),` `size_increase_decrease (info \| search)`
	AUTHOR	`Leo Crabbe`

BP1267

Any two lines intersect, and no three lines share an intersection point vs. not so.

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	COMMENTS	`Left-sorted examples divide the plane into a maximal amount of disconnected white regions by a given number of "cuts". The number of regions in one of these examples will be the nth value of of the Lazy Caterer sequence ( https://oeis.org/A000124 ), where n is the number of lines.`
	REFERENCE	`https://en.wikipedia.org/wiki/Lazy_caterer%27s_sequence`
	CROSSREFS	`Adjacent-numbered pages: BP1262 BP1263 BP1264 BP1265 BP1266 * BP1268 BP1269 BP1270 BP1271 BP1272`
	KEYWORD	`precise, allsorted, notso, perfect`
	AUTHOR	`Leo Crabbe`

BP1268

Palindromic when elements are grouped into (more than one) equal-sized blocks vs. no grouping of elements into (more than one) equal-sized blocks is palindromic.

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	COMMENTS	`Any palindrome would be sorted left, except strings of length zero or one.`
	CROSSREFS	`Adjacent-numbered pages: BP1263 BP1264 BP1265 BP1266 BP1267 * BP1269 BP1270 BP1271 BP1272 BP1273`
	KEYWORD	`precise, allsorted, unwordable, notso, sequence, traditional, miniworlds`
	CONCEPT	`element_wise_symmetry (info \| search),` `element_grouping (info \| search),` `sequence (info \| search),` `same_shape (info \| search),` `same (info \| search)`
	WORLD	`[smaller \| same \| bigger] zoom in left \| zoom in right`
	AUTHOR	`Leo Crabbe`

BP1273

Sequence contains each possible way its distinct elements can be arranged as a subsequence vs. not so.

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	REFERENCE	`https://en.wikipedia.org/wiki/Superpermutation`
	CROSSREFS	`Adjacent-numbered pages: BP1268 BP1269 BP1270 BP1271 BP1272 * BP1274 BP1275 BP1276 BP1277 BP1278`
	EXAMPLE	`There are 6 ways of arranging the letters A, B and C: ABC, ACB, BAC, BCA, CAB, and CBA. The string "ABCABACBA" contains each of these as a substring, and would therefore be sorted left.`
	KEYWORD	`precise, allsorted, notso, sequence, traditional, miniworlds`
	CONCEPT	`sequence (info \| search),` `overlap (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP1274

Reversing the sequence permutes the objects vs. not.

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	COMMENTS	`Equivalently, some permutation of the objects reverses the sequence vs. not.` `Palindromes fit left. Strings of distinct objects repeated any number of times fit left.`
	CROSSREFS	`Adjacent-numbered pages: BP1269 BP1270 BP1271 BP1272 BP1273 * BP1275 BP1276 BP1277 BP1278 BP1279`
	KEYWORD	`nice, precise, allsorted, notso, sequence, miniworlds`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP1279

Circled points are all possible vertices a square with a particular side length can take, provided that each of its corners lie on a grid point vs. not so.

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	COMMENTS	`Rotation of the square is allowed.`
	CROSSREFS	`See BP1280 for version with hexagons on a hexagonal grid.` `Adjacent-numbered pages: BP1274 BP1275 BP1276 BP1277 BP1278 * BP1280 BP1281 BP1282 BP1283 BP1284`
	KEYWORD	`hard, precise, allsorted, unwordable, hardsort, left-finite, right-finite, left-full, fixedgrid, preciseworld`
	CONCEPT	`square (info \| search)`
	AUTHOR	`Leo Crabbe`

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