Bongard Problems with solutions related to cross sections of 3D shapes.

Adjacent-numbered pages:
BP812 BP813 BP814 BP815 BP816  *  BP818 BP819 BP820 BP821 BP822

meta (see left/right), links, metaconcept

bp [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP811 BP812 BP813 BP814 BP815  *  BP817 BP818 BP819 BP820 BP821

precise, notso, stretch, unstable, perfect

fill_shapes [smaller | same | bigger]

Leo Crabbe

All examples in this Problem are position-independent Bongard Problems.

Positioning here includes objects' positions within the panels and objects' positions relative to each other.

There are very subtle distinctions to be made between the usage of variance of position in these BPs for the sake of noise (obscuring the solution eg. BP557), clarity (generalising the solution to make it more fundamental eg. BP79) or help (aiding the observer in finding the solution eg. BP334). There is certainly a degree of overlap between these three definitions, they are not disconnected.

Adjacent-numbered pages:
BP794 BP795 BP796 BP797 BP798  *  BP800 BP801 BP802 BP803 BP804

meta (see left/right), links

Leo Crabbe

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

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

Pairwise comparison Bongard Problems in which the two objects are of two fixed different types are always sorted left here. - Aaron David Fairbanks, Aug 22 2020

Adjacent-numbered pages:
BP782 BP783 BP784 BP785 BP786  *  BP788 BP789 BP790 BP791 BP792

meta (see left/right), links, keyword

Leo Crabbe

All examples in this Problem are groups of black dots.

The nth triangular number is the sum over the natural numbers from 1 to n, where n > 0. Note: 0 is the 0th triangular number. The first few triangular numbers are 0, 1, 3 (= 1+2) and 6 (= 1+2+3)

Adjacent-numbered pages:
BP564 BP565 BP566 BP567 BP568  *  BP570 BP571 BP572 BP573 BP574

nice, precise, allsorted, notso, number, math, left-narrow, left-null, help, preciseworld

dots [smaller | same | bigger]

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

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-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

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

fill_shape [smaller | same | bigger]

Leo Crabbe

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