The On-Line Encyclopedia of Bongard Problems

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Identical pairs of symbols are always on opposite sides of the cube (when assembled) vs. some identical symbols are next to each other on the cube.

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	CROSSREFS	`Adjacent-numbered pages: BP1264 BP1265 BP1266 BP1267 BP1268 * BP1270 BP1271 BP1272 BP1273 BP1274`
	KEYWORD	`left-finite, right-finite, finishedexamples, preciseworld`
	CONCEPT	`cube (info \| search),` `correspondence (info \| search),` `3d_net (info \| search),` `3d_solid (info \| search)`
	AUTHOR	`Leo Crabbe`

BP1270

Straight line can separate the two types of figures vs. not so.

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	COMMENTS	`This Bongard Problem is a sort of meta-problem. The problems inside the boxes aren't exactly Bongard problems, but they do deal with categorization. The objects must be split into two obvious and distinct groups, but one is not given any examples to work with.`
	CROSSREFS	`The two sides of BP3 are the same as the two sides in the first box.` `Adjacent-numbered pages: BP1265 BP1266 BP1267 BP1268 BP1269 * BP1271 BP1272 BP1273 BP1274 BP1275`
	KEYWORD	`notso`
	AUTHOR	`Ben`

BP1271

Positive correlation vs. negative correlation.

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	COMMENTS	`All examples in this Bongard Problem are scatter plots. Each dot represents a data point.` `"Positive correlation" means that when the X value increases, the Y value tends to increase as well (in the long run), while "negative correlation" means that when the X value increases, the Y value tends to decrease.`
	CROSSREFS	`Adjacent-numbered pages: BP1266 BP1267 BP1268 BP1269 BP1270 * BP1272 BP1273 BP1274 BP1275 BP1276`
	EXAMPLE	`Example TM4854 does not fit on either side because when the X value increases, the Y value stays the same.` `Example TM4855 does not fit on either side because there is no correlation.`
	KEYWORD	`fuzzy, minimal, unwordable, teach, spectrum, dual, handed, leftright, updown, rotate, stable, hardsort, left-narrow, right-narrow`
	AUTHOR	`Ben`

BP1272

Square minus circle vs. circle minus square.

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	COMMENTS	`All examples in this Bongard Problem are solid black shapes.`
	CROSSREFS	`BP345 is "Intersection of circle and square vs. union of circle and square."` `Adjacent-numbered pages: BP1267 BP1268 BP1269 BP1270 BP1271 * BP1273 BP1274 BP1275 BP1276 BP1277`
	EXAMPLE	`EX1 is ambiguous because it might be a small circle minus a large square, or a small square minus a large circle.`
	KEYWORD	`easy, nice, precise, dual`
	CONCEPT	`subtraction (info \| search),` `overlap (info \| search)`
	AUTHOR	`Ben`

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`

BP1275

There is a way of grouping elements into (more than one) equal-sized blocks such that no block appears twice vs. there exists no such grouping.

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	COMMENTS	`Sequences with a prime number of elements are sorted left when all their elements are unique, and sorted right otherwise.`
	CROSSREFS	`Adjacent-numbered pages: BP1270 BP1271 BP1272 BP1273 BP1274 * BP1276 BP1277 BP1278 BP1279 BP1280`
	EXAMPLE	`The sequence ABBABB would be sorted left, as it could be grouped into (AB)(BA)(BB), where each block is unique.`
	KEYWORD	`precise, unwordable, notso, sequence, miniworlds`
	CONCEPT	`element_grouping (info \| search)`
	AUTHOR	`Leo Crabbe`

BP1276

Ways of representing the sequence "ABABCBACCBAC" by grouping its elements into equal-sized blocks and relabelling them (identical blocks are represented by the same element) vs. representations of different sequences.

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	COMMENTS	`The solver isn't expected to word their solution as it is put above, but to notice that the sequences on the left all "agree" with each other under the aforementioned grouping rule. The 12-element sequence in EX10249 can be "squashed down" into the 6-element sequence in EX10251, which in turn can be "squashed down" into the 3-element sequence in EX10252.`
	CROSSREFS	`Compare to BP121` `Adjacent-numbered pages: BP1271 BP1272 BP1273 BP1274 BP1275 * BP1277 BP1278 BP1279 BP1280 BP1281`
	KEYWORD	`unwordable, notso, arbitrary, sequence`
	CONCEPT	`element_grouping (info \| search)`
	AUTHOR	`Leo Crabbe`

BP1277

Bongard Problem with solution relating to concept: element grouping vs. Bongard Problem unrelated to this concept.

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	CROSSREFS	`Adjacent-numbered pages: BP1272 BP1273 BP1274 BP1275 BP1276 * BP1278 BP1279 BP1280 BP1281 BP1282`
	KEYWORD	`meta (see left/right), metaconcept`
	CONCEPT	`This MBP is about BPs that feature concept: "element_grouping"`
	WORLD	`bp [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP1278

There is a way of dividing the grid into (more than one) equal-sized blocks such that no block appears more than once vs. there exists no such way of dividing the grid.