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BP558 Point sequence that is increasing or decreasing in height vs. point sequence that alternates in height
(edit; present; nest [left/right]; search; history)
COMMENTS

All examples in this Problem are groups of 3 dots.


Any example where 2 adjacent dots have the same height would be ambiguously sorted.

CROSSREFS

Adjacent-numbered pages:
BP553 BP554 BP555 BP556 BP557  *  BP559 BP560 BP561 BP562 BP563

EXAMPLE

Reading from right to left in the first box on the left hand side: the 2nd dot is higher than the 1st, and the 3rd is higher than the 2nd, so the sequence of dots is strictly increasing in height.

KEYWORD

nice, precise, antihuman, orderedtriplet, preciseworld

CONCEPT coordinate (info | search)

WORLD

three_points [smaller | same | bigger]

AUTHOR

Leo Crabbe

BP559 Cross section of a cube vs. not cross section of a cube
?
(edit; present; nest [left/right]; search; history)
COMMENTS

All examples are solid black shapes.


This problem is absurdly hard. It makes a good extreme example. - Aaron David Fairbanks, Nov 23 2020

CROSSREFS

Adjacent-numbered pages:
BP554 BP555 BP556 BP557 BP558  *  BP560 BP561 BP562 BP563 BP564

KEYWORD

hard, precise, allsorted, notso, stretch, challenge, left-narrow, perfect

CONCEPT cube (info | search),
cross_section (info | search)

WORLD

fill_shape [smaller | same | bigger]

AUTHOR

Leo Crabbe

BP560 There exists a closed trail that hits each edge exactly once vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

Left examples are called "Eulerian graphs."


A connected graph is Eulerian if and only if each vertex is incident to an even number of edges.

CROSSREFS

Adjacent-numbered pages:
BP555 BP556 BP557 BP558 BP559  *  BP561 BP562 BP563 BP564 BP565

KEYWORD

precise, allsorted, math, traditional, preciseworld

CONCEPT graph (info | search),
distinguishing_crossing_curves (info | search),
all (info | search),
even_odd (info | search),
existence (info | search)

WORLD

connected_graph [smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP561 Meta Bongard Problems fitting in their own world vs. other meta Bongard Problems.
BP561
(edit; present; nest [left/right]; search; history)
COMMENTS

This prompts thinking about left-self and right-self keywords.

CROSSREFS

Adjacent-numbered pages:
BP556 BP557 BP558 BP559 BP560  *  BP562 BP563 BP564 BP565 BP566

KEYWORD

meta (see left/right), links, left-self, sideless, metameta

WORLD

linksbp [smaller | same | bigger]
zoom in left (bp_in_own_world)

AUTHOR

Aaron David Fairbanks

BP562 There exists a closed trail that hits each vertex exactly once vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

Left examples are called "Hamiltonian graphs."

CROSSREFS

Adjacent-numbered pages:
BP557 BP558 BP559 BP560 BP561  *  BP563 BP564 BP565 BP566 BP567

KEYWORD

math, traditional

CONCEPT graph (info | search),
distinguishing_crossing_curves (info | search)

WORLD

connected_graph [smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP563 Bongard Problems such that there is a way of making an infinite list of all relevant possible left-sorted examples vs. Bongard Problems where there is no such way of listing all left-sorted examples.
BP386
BP394
BP904
BP922
BP926
BP931
BP956
BP997
BP1057
BP1072
BP1146
BP1147
BP1148
BP1149
BP1150
BP1197
BP1199
BP1200
BP1201
BP319
BP345
BP351
BP559
BP818
?
BP329
(edit; present; nest [left/right]; search; history)
COMMENTS

Left-sorted Problems have the keyword "left-listable" on the OEBP.


All the possible left examples for the BPs on the left side of this problem could be listed in one infinite sequence. Right examples here are Problems for which no such sequence can exist.


This depends on deciding what images should be considered "the same thing", which is subjective and context-dependent.


All examples in this Bongard Problem have an infinite left side (they do not have the keyword left-finite).


The mathematical term for a set that can be organized into an infinite list is a "countably infinite" set, as opposed to an "uncountably infinite" set.

Another related idea is a "recursively enumerable" a.k.a. "semi-decidable" set, which is a set that a computer program could list the members of.

The keyword "left-listable" is meant to be for the more general idea of a countable set, which does not have to do with computer algorithms.


Note that this is not just BP940 (right-listable) flipped.


It seems in practice, Bongard Problems that are left-listable are usually also right-listable because the whole class of relevant examples is listable. A keyword for just plain "listable" may be more useful. Or instead keywords for left- versus right- semidecidability, in the sense of computing. - Aaron David Fairbanks, Jan 10 2023

REFERENCE

https://en.wikipedia.org/wiki/Countable_set

CROSSREFS

See left-finite, which distinguishes between a finite left side and infinite left side.


"Left-listable" BPs are typically precise.

Adjacent-numbered pages:
BP558 BP559 BP560 BP561 BP562  *  BP564 BP565 BP566 BP567 BP568

KEYWORD

math, meta (see left/right), links, keyword

WORLD

bp_infinite_left_examples [smaller | same | bigger]
zoom in right (left_uncountable_bp)

AUTHOR

Leo Crabbe

BP564 Discrete points intersecting boundary of convex hull vs. connected segment intersecting boundary of convex hull
(edit; present; nest [left/right]; search; history)
COMMENTS

If a "string" is wound tightly around the shape, does one of its segments lie directly on the shape?


All examples in this Problem are connected line segments or curves.


We are taking lines here to be infinitely thin, so that if the boundary of the convex hull intersects the endpoint of a line exactly it is understood that they meet at 1 point.

CROSSREFS

Adjacent-numbered pages:
BP559 BP560 BP561 BP562 BP563  *  BP565 BP566 BP567 BP568 BP569

EXAMPLE

Imagine wrapping a string around the pointed star. This string would take the shape of the boundary of the star's convex hull (a regular pentagon), and would only touch the star at the end of each of its 5 individual tips, therefore the star belongs on the left.

KEYWORD

hard, nice, allsorted, solved, perfect

AUTHOR

Leo Crabbe

BP565 Bongard Problems that are hard for humans to solve but easier for computers to solve vs. Bongard Problems that are hard for computers to solve but easier for humans to solve.
BP112
BP558
BP941
BP1008
BP1055
BP100
BP170
BP190
BP193
BP197
BP199
BP235
BP237
BP252
BP331
BP349
BP373
BP382
BP524
BP551
BP565
BP812
BP839
BP844
BP862
BP869
BP882
BP930
BP939
BP1002
BP1004
BP1092
BP1110
BP1260
(edit; present; nest [left/right]; search; history)
COMMENTS

Left examples have the keyword "antihuman" on the OEBP.

Right examples have the keyword "anticomputer" on the OEBP.


Easy abstract Bongard Problems are typically anticomputer Bongard Problems.

CROSSREFS

See keyword help for Bongard Problems that can be made easier for humans to solve by the selection of helpful examples.

Adjacent-numbered pages:
BP560 BP561 BP562 BP563 BP564  *  BP566 BP567 BP568 BP569 BP570

KEYWORD

spectrum, anticomputer, meta (see left/right), links, keyword, right-self, viceversa

WORLD

bp [smaller | same | bigger]

AUTHOR

Leo Crabbe

BP566 Meta Bongard Problems of the form "[transformation] applied to some examples switch their sorting vs. sorting is invariant under [transformation]" vs. other meta Bongard Problems.
BP510
BP535
BP539
BP552
BP553
BP554
BP556
BP964
BP974
BP1009
BP1112
BP1140
BP1195
?
BP575
(edit; present; nest [left/right]; search; history)
COMMENTS

Left-sorted Bongard Problems have the keyword "invariance" on the OEBP.


Bongard Problems labelled "invariance" are usually (but not always) about transformations that can be undone by other transformations of the same class. (The technical term for this kind of transformation is an "isomorphism".)


When the transformations used in a "invariance" Bongard Problem vary continuously, there could usually be made a corresponding stability Bongard Problem. Stability Bongard Problems are like "invariance" Bongard Problems but for arbitrarily small applications of [transformation] affecting examples' sorting.


Potentially, stability Bongard Problems could be considered "invariance" Bongard Problems. On one hand, they are different, since checking whether arbitrarily small transformations switch an example's sorting is different from checking whether a particular transformation switches an example's sorting; the former is infinitely many conditions. On the other hand, there is actually only finitely much detail in any of the examples, and in practice a stability Bongard Problem generally just amounts to "a small application of [transformation] switches an example's sorting vs. not".

(The keyword gap is another example of a Bongard Problem currently labelled with "invariance" that arguably does not technically fit.)


Also, dependence Bongard Problems could be considered "invariance" Bongard Problems, where the relevant kind of transformation is swapping the example out for any other example that shares the relevant property.

CROSSREFS

"Invariance" Bongard Problems are notso Bongard Problems.

"Invariance" Bongard Problems are often keywords (keyword keyword) on the OEBP.


See keyword problemkiller, which is about transformations making all sorted examples unsortable.

Adjacent-numbered pages:
BP561 BP562 BP563 BP564 BP565  *  BP567 BP568 BP569 BP570 BP571

KEYWORD

meta (see left/right), links, keyword, metameta

WORLD

linksbp [smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP567 Visual Bongard Problems that would sort a blank panel on the left vs. visual Bongard Problems that would sort a blank panel on the right.
BP1
BP137
BP384
BP525
BP544
BP569
BP904
BP905
BP915
BP945
BP962
BP1056
BP1219
BP1260
BP1261
BP1262
BP211
BP334
BP364
BP541
BP542
BP902
BP953
BP959
BP966
BP988
BP989
BP1008
BP1080
BP1089
BP1093
BP1108
BP1141
BP1151
BP1161
BP1220
BP1249
BP1252
BP1263
(edit; present; nest [left/right]; search; history)
COMMENTS

Left-sorted BPs have the keyword "left-null" on the OEBP.

Right-sorted BPs have the keyword "right-null" on the OEBP.

CROSSREFS

See BP796 for the version with pictures of Bongard Problems (miniproblems) instead of links to pages on the OEBP.

See BP1160 for the version about an all-black panel instead of all-white.

Adjacent-numbered pages:
BP562 BP563 BP564 BP565 BP566  *  BP568 BP569 BP570 BP571 BP572

KEYWORD

meta (see left/right), links, keyword, side, wellfounded, testexample

WORLD

visualbp [smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

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