Adjacent-numbered pages:
BP1006 BP1007 BP1008 BP1009 BP1010  *  BP1012 BP1013 BP1014 BP1015 BP1016

hard, precise, stretch, challenge, left-narrow, perfect, preciseworld

fill_polygon [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP1033 BP1034 BP1035 BP1036 BP1037  *  BP1039 BP1040 BP1041 BP1042 BP1043

hard, precise, allsorted, handed, leftright, math, challenge, meta (see left/right), miniproblems, assumesfamiliarity, structure, preciseworld, presentationinvariant

boxes_dots_bpimage_clear_set_of_numbers [smaller | same | bigger]

Aaron David Fairbanks

Adjacent-numbered pages:
BP1035 BP1036 BP1037 BP1038 BP1039  *  BP1041 BP1042 BP1043 BP1044 BP1045

hard, precise, allsorted, convoluted, notso, handed, leftright, math, challenge, meta (see left/right), miniproblems, assumesfamiliarity, structure, preciseworld, presentationinvariant

boxes_dots_bpimage_clear_set_of_numbers [smaller | same | bigger]

Aaron David Fairbanks

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

Adjacent-numbered pages:
BP1050 BP1051 BP1052 BP1053 BP1054  *  BP1056 BP1057 BP1058 BP1059 BP1060

hard, antihuman

fill_quadrilateral [smaller | same | bigger]

Jago Collins

All examples in this Problem are grids consisting of two objects.

Adjacent-numbered pages:
BP1118 BP1119 BP1120 BP1121 BP1122  *  BP1124 BP1125 BP1126 BP1127 BP1128

EX9124 shows a 9 square by 9 square grid. Take each tile to be 3 squares by 3 squares; there is a 3 tile by 3 tile checkerboard pattern. (One of these tiles is itself a checkerboard pattern; the other is all black squares.)

hard, nice, precise, allsorted, hardsort, grid, miniworlds

Jago Collins

This was an unintended solution for BP1130.

In category theory lingo, left examples are built by repeated horizontal composition and vertical composition. (Making horizontal lines as 0-ary vertical compositions is here forbidden.)

Anything fitting right in BP1130 fits right here.

Adjacent-numbered pages:
BP1124 BP1125 BP1126 BP1127 BP1128  *  BP1130 BP1131 BP1132 BP1133 BP1134

hard, less, convoluted, solved, inductivedefinition

[smaller | same | bigger]

Aaron David Fairbanks

The description in terms of rectangles was noted by Sridhar Ramesh when he solved this.

All examples in this Bongard Problem feature arced line segments connected at endpoints; these segments do not cross across one another and they are nowhere vertical; they never double back over themselves in the horizontal direction.

Furthermore, in each example, there is a single leftmost point and a single rightmost point, and every segment is part of a path bridging between them. So, there is a topmost total path of segments and bottommost total chain of segments.

Any picture on the left can be turned into a subdivided rectangle by the process of expanding points into vertical lines.

Here is another answer:

"Right examples: some junction point has a single line coming out from either the left or right side."

If there is some junction point with only a single line coming out from a particular side, the point cannot be expanded into a vertical segment with two horizontal segments bookending its top and bottom (as it would be if this were a subdivision of a rectangle).

And this was the original, more convoluted idea of the author:

"Start with a string along the top path. Sweep it down, region-by-region, until it lies along the bottom path. The string may only enter a region when it fully covers that region's top edge and likewise it must exit by fully covering the bottom edge. Only in left images can this process be done so that no segment of the string ever hesitates."

Quite convoluted when spelled out in detail, but not terribly complicated to imagine visually. (See the keyword unwordable.)

The string-sweeping answer is the same as the rectangle answer because a rectangle represents the animation of a string throughout an interval of time. (A horizontal cross-section of the rectangle represents the string, and the vertical position is time.) Distorting the rectangle into a new shape is the same as animating a string sweeping across that new shape.

In particular, shrinking vertical lines of a rectangle into points means just those points of the string stay still as the string sweeps down.

The principle that horizontal lines subdividing the original rectangle become the segments in the final picture corresponds to the idea that the string must enter or exit a single region all at once.

BP1129 started as an incorrect solution for this Bongard Problem. Anything fitting right in BP1130 fits right in BP1129.

Adjacent-numbered pages:
BP1125 BP1126 BP1127 BP1128 BP1129  *  BP1131 BP1132 BP1133 BP1134 BP1135

hard, unwordable, solved

[smaller | same | bigger]

Aaron David Fairbanks

Another wording: "can be repeatedly broken along 'fault lines' to yield individual pieces vs not."

Robert Dawson, A forbidden suborder characterization of binarily composable diagrams in double categories, Theory and Applications of Categories, Vol. 1, No. 7, p. 146-145, 1995.

All of the examples fitting left here would fit right in BP1199 except for (1) a single rectangle, (2) two rectangles stacked vertically, or (3) two rectangles side by side horizontally.

All of the examples fitting right in in BP1097 (re-styled) would fit right here (besides a single solid block, but that isn't shown there).

Adjacent-numbered pages:
BP1195 BP1196 BP1197 BP1198 BP1199  *  BP1201 BP1202 BP1203 BP1204 BP1205

hard, precise, challenge, proofsrequired, inductivedefinition, left-listable, right-listable

Aaron David Fairbanks

https://en.wikipedia.org/wiki/Hex_(board_game)

Adjacent-numbered pages:
BP1240 BP1241 BP1242 BP1243 BP1244  *  BP1246 BP1247 BP1248 BP1249 BP1250

hard, precise, convoluted, dual, rotate, boundingbox, hardsort, challenge, proofsrequired, bordercontent

Aaron David Fairbanks