Panels on the left hand side contain geometrical objects that appear distorted (to a human) due to surrounding information.

Vicente Sierra-Vázquez & Ignacio Serrano-Pedraza, Application of Riesz transforms to the isotropic AM-PM decomposition of geometrical-optical illusion images, April 2010, Figure 1.

Adjacent-numbered pages:
BP934 BP935 BP936 BP937 BP938  *  BP940 BP941 BP942 BP943 BP944

fuzzy, anticomputer, subjective, contributepairs, invalid, experimental

Leo Crabbe

Adjacent-numbered pages:
BP933 BP934 BP935 BP936 BP937  *  BP939 BP940 BP941 BP942 BP943

meta (see left/right), links, metaconcept

bp [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP932 BP933 BP934 BP935 BP936  *  BP938 BP939 BP940 BP941 BP942

precise, allsorted, unstable, left-narrow, perfect, unorderedpair

2_fill_shapes [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP931 BP932 BP933 BP934 BP935  *  BP937 BP938 BP939 BP940 BP941

meta (see left/right), links, metaconcept

bp [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP930 BP931 BP932 BP933 BP934  *  BP936 BP937 BP938 BP939 BP940

nice, precise, allsorted, unstable, left-narrow, perfect, pixelperfect, unorderedpair

2_fill_shapes [smaller | same | bigger]

Leo Crabbe

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

Strictly this Problem's solution is not actually about gravity, it is about a constant downwards force (the ball's time-independent path does not depend on the magnitude of the force, only direction). The phrasing for the solution is a shorthand that takes advantage of human physical intuition.

Adjacent-numbered pages:
BP928 BP929 BP930 BP931 BP932  *  BP934 BP935 BP936 BP937 BP938

physics

dot_with_lines_or_curves [smaller | same | bigger]

Leo Crabbe

Complete graphs with zero, one, two, or three vertices would be ambiguously categorized (fit in overlap of both sides).

Left examples are called "fully connected graphs." Right examples are called "cycle graphs."

Adjacent-numbered pages:
BP927 BP928 BP929 BP930 BP931  *  BP933 BP934 BP935 BP936 BP937

precise, left-narrow, right-narrow, both, preciseworld

connected_graph [smaller | same | bigger]

Aaron David Fairbanks

Right examples are called "derangements".

Adjacent-numbered pages:
BP926 BP927 BP928 BP929 BP930  *  BP932 BP933 BP934 BP935 BP936

handed, leftright, sequence, traditional, left-listable, right-listable

dot_clusters_sequence_horizontal [smaller | same | bigger]

Aaron David Fairbanks

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

BPs should be marked "help" when the OEBP wants most examples (at least on one side) to be helpful (not when just one or two uploaded examples are helpful).

Helpfulness can be a spectrum; most Bongard Problems are helpful to some degree just by not using the most convoluted unintelligible examples possible.

Examples that are helpful to people are often not particularly helpful to computers.

Any helpful Bongard Problem has a harder, not helpful version. For example, BP384 (square number of dots versus non-square number of dots) would be much harder if all examples had hundreds of dots that weren't arranged recognizably. Instead, the dots in the examples are always arranged in shapes that make the square-ness or non-square-ness of the numbers easy to check without brute counting.

When all examples in a Bongard Problem are helpful, it may become unclear whether the helpfulness is part of the Bongard Problem's solution.

E.g.: Is the left-hand side of BP384 "square number of dots", or is it "square number of dots that are arranged in a helpful way so as to communicate the square-ness"?

See seemslike, where examples being helpful is an irremovable aspect of the Bongard Problem's solution.

Adjacent-numbered pages:
BP925 BP926 BP927 BP928 BP929  *  BP931 BP932 BP933 BP934 BP935

anticomputer, meta (see left/right), links, keyword, oebp, instruction

bppage [smaller | same | bigger]
zoom in left (help_bp)

Aaron David Fairbanks