"Full overlapping not allowed" means you cannot overlay an image onto itself without moving it; if this were allowed all images would be sorted on the left. The copies can be moved around (translated) in 2D but can not be flipped or rotated.

There are examples on the right drawn with thick lines, and these could be created by copying an image with slightly thinner lines and moving it over a tiny amount. If you fix this issue by saying "the copy has to be moved over more than a tiny amount" then the Bongard Problem is perfect but not precise, but if you fix this issue by saying "interpret the figures as made up of (infinitesimally) thin lines" then it's precise but not perfect. - Aaron David Fairbanks, Jun 17 2023

Adjacent-numbered pages:
BP941 BP942 BP943 BP944 BP945  *  BP947 BP948 BP949 BP950 BP951

nice, notso, creativeexamples

Leo Crabbe

Adjacent-numbered pages:
BP940 BP941 BP942 BP943 BP944  *  BP946 BP947 BP948 BP949 BP950

precise, allsorted, number, left-null, help, preciseworld

dots [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP937 BP938 BP939 BP940 BP941  *  BP943 BP944 BP945 BP946 BP947

precise, stretch, boundingbox, invalid, experimental, preciseworld

rectangular_arrangement_of_white_pixels [smaller | same | bigger]
zoom in left (square_arrangement_of_white_pixels) | zoom in right (oblong_rectangular_arrangement_of_white_pixels)

Leo Crabbe

Evidently this solution is stretching the idea of what makes images "different". Each corresponding pair of panels would be parsed as identical in any other case. It is worth noting that if you view this Problem on a template, the solution no longer applies at all.

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BP936 BP937 BP938 BP939 BP940  *  BP942 BP943 BP944 BP945 BP946

less, precise, dual, antihuman, contributepairs, invalid, experimental, funny

Leo Crabbe

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.

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BP934 BP935 BP936 BP937 BP938  *  BP940 BP941 BP942 BP943 BP944

fuzzy, anticomputer, subjective, contributepairs, invalid, experimental

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

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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.

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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.

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BP928 BP929 BP930 BP931 BP932  *  BP934 BP935 BP936 BP937 BP938

physics

dot_with_lines_or_curves [smaller | same | bigger]

Leo Crabbe

The mathematical terms for these operations are even and odd permutations.

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precise, allsorted, math, left-narrow, right-narrow, unorderedpair, preciseworld, left-listable

Leo Crabbe