Index of Bongard Problems

This index contains links to all Bongard problems available to the author. If the reader wants to make their own contribution and enrich this collection, please read this link: Design Your Own Bongard Problems, and also this one: Things Not Allowed in Bongard Problems (but with a grain of salt — use your own judgment).

List of problems by incremental number

There is also a number of meta-Bongard Problems (and meta-meta-BP’s) that the author and Joseph Insana, in cooperation, came up with. What is meant by the term “meta-BP” can be understood by looking at problem #200, above. The best description of meta-BP’s on the web can be found under these pages by Joseph Insana. Implemented meta-BP’s exist also in Aaron David Fairbanks’ collection.

Concepts that Appear in Solutions of Bongard Problems

The following can be seen as a “hints” list, in case you don’t want to be given directly the solutions of BP’s (which are here), but only to know what concepts the solution involves. The concepts index also allows designers to find out if a BP already exists that implements the idea they came up with.

Color-coding showing which concept was originally introduced by which designer (listed chronologically):

Notes:

1. Those among the concepts listed below that are primitive are marked with this symbol: on the leftmost column. A composite (non-primitive) concept, for example, is “equilateral triangle”, which can be decomposed based on the primitive concepts: “Vertex of two lines that meet”, “Angle”, “Length of lines”, “Sameness based on feature”, and “Number”. Note: some concepts, such as “Equality”, “Zero” (or any specific small number), “Barycenter of object”, and several more, are too ubiquitous to be included in this list.

2. Sometimes a Bongard Problem (BP) is listed next to a concept not because the concept appears directly in the solution of that BP, but because it is used indirectly. If the concept is present (can be seen) in the BP in general, but is not used (directly or indirectly) in the solution, the BP is not listed next to the concept.

3. This list should be useful to both creators of new BP’s and those few programmers who attempt to automate the solution of BP’s (as the author has done in the past). Thus, creators of new BP’s might want to check which already-known BP’s use the concepts (primitive or composite) in their new BP, to see if their BP is unique. Programmers might want to view this list in the spirit of: “My program should be able to handle these concepts, particularly the primitive ones”.

4. Lastly, a third possible category of people who might find this list useful are those who, like the author, are interested not simply in the automation of solving BP’s, but in modeling visual cognition. This list then can be seen as an attempt to answer the question: which concepts are primitive () in 2-D, black-and-white, visual cognition? Since the list has been constructed by many people contributing to it, it can be viewed as the end-product of the indirect and uncoordinated effort of many people who have created BP’s over a period of many years. It is thus very interesting to see if anyone can find some new primitive concept in the future.

If you see any corrections that need to be made, but adhering to the spirit of this index (see Notes, above), please notify me.

	Concept	Bongard Problems with that concept in their solution
	Above / Below	36, 37, 48, 67, 68, 337
	Absence seen as presence	121, 193, 297
	Absolute location in the box	8, 109, 207, 298
	Acceleration / Deceleration	300, 357, 392
	Acute angle	17, 32, 33, 292
	Addition (arithmetic)	230, 233, 381
	Adjacent	142
	All ( > 2, within the box) / Not all	22, 39, 56, 78, 92, 108, 120, 133, 134, 161, 177, 189, 190, 208, 219, 244, 260, 296, 368, 376, 388
	Analogy (a : b :: c : d)	361, 362
	Angle	17, 32, 33, 77, 262, 328, 329
	Approximately	see: Imperfection (small)
	Arrow	52, 102, 118, 358, 363
	Average (of values of some feature)	22, 48
	Background of box	157
	Background of shape	194
	Barycenter of cluster	310
	Base of object	116, 132, 199, 273
	Becoming larger / Becoming smaller	43, 74, 108, 179, 180, 278, 286, 300, 318, 341, 350, 351, 352, 357, 365
	Belongs	see: Categorization
	Between	44, 94, 102, 111, 117
	Bongard Problem	200
	Box (of Bongard Problem)	8, 157, 209, 225, 231, 234, 257, 321
	Branching point on linear structure	67, 68, 86, 92, 93, 113
	Categorization	346, 349, 372, 393
	Center of box of circle of mass	209 104, 117, 133, 134, 178 366, 367
	Chess-like board and concepts	376
	Clockwise / Counterclockwise turn	16, 54, 120, 208, 226, 259, 313, 336, 363
	Close to / Far from (spatially)	see: Near / Far (spatially)
	Closedness / Openness	15, 264, 313
	Clustering (within the box) based on feature value(s) based on number based on relation based on shape needs cluster of 1	25, 26, 27, 41, 48, 49, 58, 81, 89, 90, 141, 142, 143, 144, 145, 147, 149, 156, 166, 167, 169, 189, 205, 220, 255, 261, 307, 308, 349, 356, 364, 372 349 29, 42, 49, 61, 66, 128, 143, 144, 190, 310, 347, 349, 356 28, 58, 65, 83, 84, 99, 141, 142, 143, 144, 145, 169, 186, 187, 205, 253, 349 81, 83, 89, 90, 149, 156, 166, 167, 169, 220, 347, 349
	Coincidence of 3+ lines at a point	78, 86, 162, 287
	Collinearity (straight line or curve)	40, 41, 42, 64, 66, 92, 93, 94, 96, 99, 133, 134, 147, 161, 212, 254, 285, 302
	Color (as in: outlined, filled)	see: Outlined / Filled
	Completeness	379
	Concave / Convex region of a curve	75, 215, 274
	Concavity on object	see: Indentation
	Concavity / Convexity	4, 136, 174, 177, 182
	Conjunction (logical)	26, 28, 140, 141, 142, 143, 144, 145, 325
	Continuous change	see: Discrete / Continuous quantity
	Contrast high / low	225
	Convex hull	12, 65, 81, 82, 83, 84, 97, 138, 192, 242, 268, 272, 278, 280, 348
	Comparison of numbers	see: Number (of feature or object); Sameness based on number
	Completed out of the box	193, 210, 312
	Coordinate (X/Y) projected on box	8, 112
	Corner of box	243
	Correspondence	121, 151, 295
	Counting	see: Iteration (discrete, within box)
	Crossings of lines (X-points)	30, 99, 113, 114, 199, 213, 240, 261, 266, 273
	Curves / Straight lines	5, 130, 154, 155, 246, 247, 263
	Cusp	32, 44
	Cyclic path	118, 316
	Decreasing / Increasing	see: Becoming larger / Becoming smaller
	Dent	see: Indentation
	Diametrically opposite	224
	Difference (arithmetic)	164, 187, 244, 311
	Dimensionality	330, 331
	Direction (absolute, w.r.t. the box) of straight line of tangent to curve of rotation	199, 215, 234, 238, 273, 274 206 16, 54, 55, 208, 226
	Discrete / Continuous quantity	351
	Disjunction (logical)	13, 266, 270, 341
	Distractor	see: Noise
	Division of integers / Fractions	387
	Downward / Upward	179, 199, 206, 215, 234, 273, 274, 286, 337
	Elongatedness / Roundness	11, 12, 65, 168, 192, 268
	Empty / Nonempty	1, 260
	Ending of curve (tangent at its end)	72, 105, 206, 263
	Endpoints of a line or curve	62, 68, 69, 94, 96, 103, 153, 154, 198, 206, 207, 216, 230, 239, 240, 276
	Entrance / Exit	175, 191
	Entropy high / low	see: Pattern / Randomness
	Equality	see: Sameness (within the box)
	Equivalence relation	see: If and only if
	Even / Odd	150, 202, 267, 334
	Every other	162, 336
	Exclusive or operator (logical)	325
	Existence	1, 24, 26, 92, 118, 119, 131, 138, 152, 209, 210, 221, 266, 276, 290, 296, 298, 343, 347, 349, 368, 391, 394
	Exists exactly one	127, 342
	Exponential / Linear	392
	Filled / Outlined	see: Outlined / Filled
	Finite / Infinite	380
	Fractal	355, 356
	Fraction	see: Ratio
	Front / Back (in 3-D)	45, 46, 195
	Gear	336
	Ghost figure	193, 297
	Half	284
	Hole	34, 35
	Horizontal / Vertical	7, 13, 19, 65, 66, 95, 116, 132, 147, 192, 206, 268, 270, 280, 281
	Identical objects (within the box)	57, 58
	If and only if	146, 227, 290
	Imaginary entity point   line or curve   shape	52, 78, 102, 104, 117, 133, 134, 161, 162, 178, 183, 184, 249, 250, 257, 287, 310, 312, 368 64, 73, 74, 78, 81, 103, 106, 112, 117, 131, 154, 162, 163, 183, 199, 235, 238, 239, 240, 245, 250, 257, 261, 273, 275, 280, 287, 292, 312, 342, 358, 368, 383, 388, 394 96, 97, 101, 138, 193, 194, 344, 345, 362, 390, 391
	Imperfection (small)	119, 130, 148, 160, 168, 222, 223, 224, 299, 338
	Increasing / Decreasing	see: Becoming larger / Becoming smaller
	Indentation	17, 55, 76, 91, 101, 125, 129, 175, 181, 231
	Infinite 2-D plane	335
	Infinity	see: Finite / Infinite
	Innermost / Outermost	115
	Inside (see also: Outside)	29, 47, 49, 53, 71, 83, 111, 128, 146, 198, 248, 249, 291, 306, 310, 317, 367
	Interior / Exterior (of a closed area)	29, 42, 49, 84, 122, 128, 291, 306, 310
	Intersection	see: Overlapping; also: Crossings of lines (X-points)
	Intersection of sets	see: Union / Intersection (of sets)
	Irrelevant objects or features	see: Noise
	Iteration (discrete, within the box)	318, 350, 351, 352, 353, 355, 356
	Jigsaw puzzle	323
	Knot	382, 389
	Large / Small	see: Size of objects or features
	Left / Right	8, 63, 67, 68, 191, 234, 279, 280, 339
	Leftward / Rightward	43, 206, 278
	Length of lines or curves	14, 79, 80, 103, 112, 131, 151, 155, 176, 235, 237, 239, 241, 243, 256, 257, 279, 280, 314, 328
	Light / Dark	196, 236
	Loop / no loop	15, 353, 354
	Lying on (touching) a line or curve	44, 55, 66, 69, 91, 104, 117, 190, 336
	Maximum / Minimum (local)	309
	Maze	294, 314, 315, 316
	Meeting at a point after tracing path	52, 102, 358
	Meta-solution	1, 200
	Middle point of two points	112, 117, 161, 257
	Missing part / Extra part	148
	Motion needing to be imagined real	175, 201, 234, 239, 323, 336, 358, 369, 370, 376, 389 300
	Multiple (arithmetic)	319
	Near / Far (spatially)	20, 49, 51, 62, 79, 149, 151, 182, 212, 227, 243, 256, 310
	Neck, narrowing in area	18, 19, 20
	Noise	10, 25, 37, 40, 42, 58, 60, 64, 65, 66, 73, 82, 98, 105, 106, 109, 116, 123, 127, 128, 130, 131, 132, 135, 136, 162, 165, 174, 181, 183, 192, 201, 203, 204, 206, 207, 210, 222, 223, 231, 237, 242, 261, 298, 303
	Notch	see: Indentation
	Number (countable quantity of feature or object)	6, 10, 23, 27, 28, 29, 31, 53, 85, 86, 87, 88, 89, 90, 91, 96, 98, 107, 110, 114, 122, 126, 127, 141, 143, 144, 145, 151, 156, 159, 160, 164, 166, 167, 169, 181, 185, 187, 193, 194, 204, 209, 220, 231, 232, 233, 246, 247, 250, 262, 272, 277, 281, 296, 311, 318, 319
	Numerosity (sense of num. quantity)	320
	Odd	see: Even / Odd
	Order / Disorder	see: Pattern / Randomness
	Ordinal position	302, 318
	Outer outline of shape or region	322, 341
	Outlined / Filled	3, 25, 26, 27, 28, 34, 35, 41, 45, 48, 56, 58, 79, 90, 93, 94, 128, 157, 180, 189, 190, 211, 244, 261, 282, 283, 284, 288, 301, 337, 341
	Overlapping of two objects of line and object	45, 46, 81, 259, 301 163, 275
	Outside (see also: Inside)	84, 138
	Parallel / Perpendicular	35, 39, 72, 73, 91, 101, 105, 131, 165, 178, 219, 238, 239, 255
	Path	115, 118, 176, 314, 316, 321
	Pattern / Randomness	173, 359, 360
	Percent	see: Ratio
	Point (dot, no discernible shape)	166, 167, 230, 317, 318, 319
	Pointing to	52, 117
	Prime number	203, 253
	Proportional magnitudes	317, 319
	Protrusion	32, 91, 108, 125, 129, 226
	Ratio	288, 306
	Reachable / Unreachable	115, 235, 294, 315, 321
	Recursion	70, 71, 167, 186
	Regions on two sides of a curve	67, 75, 135
	Regular change / Irregular change	43, 139, 179, 218, 278, 286
	Remove (mentally) some object(s)	151
	Rotation required	175, 201, 228, 229, 259, 305
	Sameness (i.e., equality) (within the box) of feature of number of shape of concept	56, 77, 80, 103, 112, 124, 189, 190, 238, 239, 257, 328, 329 61, 128, 137, 169, 244, 281 57, 58, 125, 140, 141, 142, 143, 145, 195, 259, 293, 303, 305, 324, 338, 343 378
	Self-reference	390
	Semiplanes of a straight line	61, 81, 121, 183
	Separable / Entangled	81, 144
	Separation of objects joined together	20, 44, 55, 88, 90, 91, 94, 126, 135, 202, 309
	Sequence (arithmetic)	318
	Shape	6, 10, 13, 24, 25, 26, 28, 29, 36, 37, 38, 46, 47, 54, 55, 57, 58, 61, 64, 73, 75, 79, 80, 82, 83, 84, 97, 98, 109, 110, 111, 119, 121, 123, 125, 127, 131, 139, 146, 149, 150, 151, 153, 154, 159, 170, 187, 188, 193, 194, 242, 245, 248, 249, 272, 283, 289, 297, 298, 323
	Similarity (but not equality) (within the box) of feature of number of shape of representation	65 320 59, 60, 123, 139, 145, 188, 205, 248, 249, 282, 338 377
	Simple / Complex representation	374, 375
	Size of objects or features	2, 21, 22, 34, 38, 76, 126, 140, 211, 248, 256, 286, 295, 301
	Slope of straight line	7, 13, 39, 106, 158
	Slope positive / negative	106
	Specificity of shape (topological) of style of value of a feature	100, 170, 175 153, 154, 155, 197, 214, 228 158, 229
	Speed (or velocity)	300, 358
	Square (as opposed to rectangle)	242
	Square of a number	384
	Sum (arithmetic)	see: Addition (arithmetic)
	Symmetry axis of radial	50, 152, 258, 265, 342 172, 269
	Tangent at point of curve	183, 238, 326, 327
	Tessellation	335
	Texture as in: outlined / filled of line or curve of area	see: Outlined / Filled 9, 63, 107, 184, 185, 212, 216, 217, 221, 248, 249 95, 196, 236
	Tiling	122, 201, 229, 283, 289, 323, 344, 386
	Together	see: Near / Far (spatially)
	Topological transformation	369, 370, 382, 389, 390
	Touchings of lines (T-points)	113, 266
	Tracing along a line or curve	16, 43, 52, 55, 67, 68, 69, 70, 72, 74, 102, 105, 115, 120, 179, 180, 198, 208, 278, 336, 340, 341, 350, 351, 352, 353, 354, 355, 356, 357, 363, 365
	Triangle equilateral incenter isosceles orthocenter used as arrow to point to	82, 151 171 80, 103 171 117, 287
	True, correct / False, incorrect	393
	Trunk of a tree-like structure	69, 70
	Tumbles/ Stays put	199, 273
	Turning direction	120
	Union / Intersection (of sets)	339, 345, 373
	Uppermost / Lowermost	116, 179
	Variance (statistical notion)	173
	Vertex of two lines that meet	116, 132, 230, 254, 307, 312, 340
	Visual disparity	see: 3-D Depth
	Visual illusion	237
	Wave	381
	Wrapping around the box	244
	Written in single-stroke	271, 340
	3-D Depth	195
	3-D Impossible solid / Possible solid	252
	3-D Net	371, 385
	3-D Solid	250, 251, 370, 371