All examples in this Problem are groups of 3 dots.

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

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

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.

nice, precise, antihuman, orderedtriplet, preciseworld

three_points [smaller | same | bigger]

Leo Crabbe

All examples are solid black shapes.

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

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

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

fill_shape [smaller | same | bigger]

Leo Crabbe

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.

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

precise, allsorted, math, traditional, preciseworld

connected_graph [smaller | same | bigger]

Aaron David Fairbanks

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

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

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

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

Aaron David Fairbanks

Left examples are called "Hamiltonian graphs."

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

math, traditional

connected_graph [smaller | same | bigger]

Aaron David Fairbanks

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

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

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

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

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

Leo Crabbe

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.

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

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.

hard, nice, allsorted, solved, perfect

Leo Crabbe

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.

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

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

bp [smaller | same | bigger]

Leo Crabbe

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.

"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

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

linksbp [smaller | same | bigger]

Aaron David Fairbanks

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

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

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

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

visualbp [smaller | same | bigger]

Aaron David Fairbanks