Numbers of dots on the left can be obtained by repeatedly doubling 1 dot.

Numbers of dots on the left are the number of corners of a cube in some dimension.

Adjacent-numbered pages:
BP983 BP984 BP985 BP986 BP987  *  BP989 BP990 BP991 BP992 BP993

stub, precise, allsorted, number, left-narrow, right-null, help, preciseworld

dots [smaller | same | bigger]

Aaron David Fairbanks

Zero is intentionally left out to avoid confusion (although it would fit right).

Adjacent-numbered pages:
BP984 BP985 BP986 BP987 BP988  *  BP990 BP991 BP992 BP993 BP994

stub, precise, number, math, left-narrow, right-null, help, preciseworld

dots [smaller | same | bigger]

Aaron David Fairbanks

Another way of thinking about the solution is considering whether a light source placed at the center of mass of a given example would illuminate the whole shape.

Every left for this Problem would be will be a left example for both BP367 and BP368.

Adjacent-numbered pages:
BP985 BP986 BP987 BP988 BP989  *  BP991 BP992 BP993 BP994 BP995

convoluted, perfect

fill_shape_seeing_point_center_of_mass_inside [smaller | same | bigger]

Leo Crabbe

This is a generalization of BP820.

Adjacent-numbered pages:
BP986 BP987 BP988 BP989 BP990  *  BP992 BP993 BP994 BP995 BP996

precise, allsorted, perfect

fill_shape [smaller | same | bigger]
zoom in left

Leo Crabbe

All examples in this Problem are solid concave black shapes. In this Problem, the "cavities" of a concave shape are defined to be the convex hull of the shape minus the shape itself. For example, if you take a bite out of the edge of a piece of paper, the piece of paper in your mouth is the cavity of the bitten piece of paper. The idea may be indefinitely extended, considering whether the cavities of the cavities are concave or convex, and so on.

Adjacent-numbered pages:
BP987 BP988 BP989 BP990 BP991  *  BP993 BP994 BP995 BP996 BP997

nice, precise, perfect, traditional

concave_fill_shape [smaller | same | bigger]

Jago Collins

Right-sorted examples are called common nets.

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

Adjacent-numbered pages:
BP988 BP989 BP990 BP991 BP992  *  BP994 BP995 BP996 BP997 BP998

stub, precise, 3d, perfect, preciseworld

polyhedron_net [smaller | same | bigger]
zoom in left (polyhedron_net_unique_solid)

Leo Crabbe

More specifically these solids are polyhedra, and are often called "space-filling".

There is ambiguity here regarding some nets that can be folded to make multiple different solids. For example EX8175 could correspond to a cuboid with a pyramid-like protrusion at each end, a protrusion at one end and an indent at the other, or 2 indents. Only the second of these options can tessellate 3D space. For clarity's sake examples like this are not sorted on either side.

Adjacent-numbered pages:
BP989 BP990 BP991 BP992 BP993  *  BP995 BP996 BP997 BP998 BP999

stub, precise, 3d, perfect, preciseworld

polyhedron_net [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP991 BP992 BP993 BP994 BP995  *  BP997 BP998 BP999 BP1000 BP1001

precise, 3d, perfect, preciseworld

polyhedron_net_unique_solid [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP992 BP993 BP994 BP995 BP996  *  BP998 BP999 BP1000 BP1001 BP1002

precise, allsorted, grid, preciseworld, left-listable, right-listable

James Tanton

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually becoming ___"; gradually becoming thickly outlined.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually becoming "gradually becoming ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and call it "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more belabored way of putting it:

Call something like "is star-shaped" a "rule". An object can satisfy a rule.

Call something like "all but one are ___" a "rule-parametrized rule". A collection of objects can satisfy a rule-parametrized rule with respect to a particular rule.

On the right: every collection fits the same rule-parametrized rule (with respect to various rules); furthermore the collection of collections fits that same rule-parametrized rule (with respect to some unrelated rule that collections can satisfy).

On the left: The collection of collections fits a rule-parametrized rule with respect to the rule of fitting that rule-parametrized rule (with respect to various rules).

Previously, an unintended solution to this BP was "not all groups share some noticeable property vs. all do." It is hard to come up with examples foiling this alternative solution because the rule-parametrized rule (see explanation above) usually has to do with not all objects in the collection fitting the rule. (See BP568, which is about BP ideas that are always overridden by a simpler solution.) The example EX10108 "all five are 'all five are ___'" was added, foiling the alternative solution.

The right side of this Problem is a subset of BP999^left.

Adjacent-numbered pages:
BP993 BP994 BP995 BP996 BP997  *  BP999 BP1000 BP1001 BP1002 BP1003

"Odd one out with respect to what property is the odd one out" would not fit left: even though this example does seem doubly-meta, it is not doubly-meta in the right way. There is no odd one out with respect to the property of having an odd one out.

Similarly, consider "gradual transition with respect to what the gradual transition is between", etc. Instead of having the form "X 'X __' ", this is more like "X [the __ appearing in 'X __']". Examples like these two could make for a different Bongard Problem.

hard, unwordable, challenge, overriddensolution, infodense, contributepairs, funny, rules, miniworlds

zoom in right

Aaron David Fairbanks