Adjacent-numbered pages:
BP389 BP390 BP391 BP392 BP393  *  BP395 BP396 BP397 BP398 BP399

hard, nice, solved, traditional, dithering, left-listable, right-listable

Jago Collins

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

Rhetorical question: Where does this Bongard Problem sort an image of itself?

See BP999 and BP1004 for similar paradoxes.

See BP517 for the version with links to pages on the OEBP instead of images of Bongard Problems (miniproblems).

Adjacent-numbered pages:
BP788 BP789 BP790 BP791 BP792  *  BP794 BP795 BP796 BP797 BP798

hard, nice, abstract, dual, handed, leftright, solved, meta (see left/right), miniproblems, creativeexamples, presentationmatters, assumesfamiliarity, structure, experimental

boxes_bpimage_sorts_self [smaller | same | bigger]
zoom in left (boxes_bpimage_sorts_self_left) | zoom in right (boxes_bpimage_sorts_self_right)

Aaron David Fairbanks

This is a very inexact definition. Some left examples arguably should be placed on the right, since the particular way they are represented is arbitrary--the Platonic solids EX6730 and primes EX6734 especially, as these show arbitrary placement and arrangement of objects. Furthermore if arbitrary representations are allowed one cannot be sure for example the right hand drawing of random numbers EX6740 does not represent "numbers" in general. Still this Bongard Problem has been solved by people.

Adjacent-numbered pages:
BP808 BP809 BP810 BP811 BP812  *  BP814 BP815 BP816 BP817 BP818

fuzzy, abstract, stretch, math, solved, collective, experimental, dithering

Aaron David Fairbanks

This is solvable; it was solved by Sridhar Ramesh.

A full turn is considered "the same angle" as no turns; likewise for adding and subtracting full turns from any angle. All sequences of angles shown start at the rightmost tick.

It doesn't matter whether the angle is measured clockwise or counterclockwise, as long as the choice is consistent.

Adjacent-numbered pages:
BP820 BP821 BP822 BP823 BP824  *  BP826 BP827 BP828 BP829 BP830

hard, convoluted, notso, math, solved

Aaron David Fairbanks

Adjacent-numbered pages:
BP844 BP845 BP846 BP847 BP848  *  BP850 BP851 BP852 BP853 BP854

hard, convoluted, solved, rules

constant_change_seq_loop_right [smaller | same | bigger]

Aaron David Fairbanks

The rows in the panels on the right hand side show all the ways you can divide a certain number of dots between a certain number of bins, ignoring which bins they are placed in.

Adjacent-numbered pages:
BP899 BP900 BP901 BP902 BP903  *  BP905 BP906 BP907 BP908 BP909

solved, left-null, grid, left-listable, right-listable

Molly C Klenzak

All examples here are in the conventional format, i.e. white background, black vertical dividing line, and examples in boxes on either side.

Border cases are Bongard Problems that always self-categorize one way given their particular visual format (e.g. fixed number of boxes), but self-categorize a different way in another slightly different format.

See BP793 "categorizes self left vs. categorizes self right".

Adjacent-numbered pages:
BP922 BP923 BP924 BP925 BP926  *  BP928 BP929 BP930 BP931 BP932

hard, solved, presentationinvariant, visualimagination

boxes_bpimage_sorts_self [smaller | same | bigger]
zoom in left (boxes_bpimage_sorts_self_incarnation_dependent) | zoom in right

Aaron David Fairbanks

In other words, we take the distance between points (a,b) and (c,d) to be equal to |c-a| + |d-b|, or, in other words, the distance of the shortest path between points that travels along grid lines. In mathematics, this way of measuring distance is called the 'taxicab' or 'Manhattan' metric. The points on the left hand side form equilateral triangles in this metric.

⠀

An alternate (albeit more convoluted) solution that someone may arrive at for this Problem is as follows: The triangles formed by the points on the left have some two points diagonal to each other (in the sense of bishops in chess), and considering the corresponding edge as their base, they also have an equal height. However, this was proven to be equivalent to the Manhattan distance answer by Sridhar Ramesh. Here is the proof:

⠀

An equilateral triangle amounts to points A, B, and C such that B and C lie on a circle of some radius centered at A, and the chord from B to C is as long as this radius.

A Manhattan circle of radius R is a turned square, ♢, where the Manhattan distance between any two points on opposite sides is 2R, and the Manhattan distance between any two points on adjacent sides is the larger distance from one of those points to the corner connecting those sides. Thus, to get two of these points to have Manhattan distance R, one of them must be a midpoint of one side of the ♢ (thus, bishop-diagonal from its center) and the other can then be any point on an adjacent side of the ♢ making an acute triangle with the aforementioned midpoint and center.

Adjacent-numbered pages:
BP929 BP930 BP931 BP932 BP933  *  BP935 BP936 BP937 BP938 BP939

hard, allsorted, solved, left-finite, right-finite, perfect, pixelperfect, unorderedtriplet, finishedexamples

3_dots_on_square_grid [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP950 BP951 BP952 BP953 BP954  *  BP956 BP957 BP958 BP959 BP960

abstract, dual, handed, leftright, solved, meta (see left/right), miniproblems, creativeexamples, assumesfamiliarity, structure, experimental

oblong_boxes_bpimage_sorts_both_sides_skewed [smaller | same | bigger]
zoom in left (oblong_boxes_bpimage_sorts_both_sides_left) | zoom in right (oblong_boxes_bpimage_sorts_both_sides_right)

Leo Crabbe