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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240
241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260
261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280
281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320
321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340
341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380
381 382 383 384 385 386 387 388 389 390 391 392 393 394            
Designers,
& color-coding:
# of
prob.
Mikhail M. Bongard 100
Douglas R. Hofstadter 56
Harry E. Foundalis 47
Joseph A. L. Insana 32
Peter Shanahan 5
Matthew J. Howells 5
Carla Rispoli 3
Andreas Gunnarsson 13
Michael Ihde 1
Pablo Barenbaum 2
Merse E. Gáspár 17
Yoon, Sang-Joon 3
“Lewis” 26
Jakub Štepo 6
Aaron David Fairbanks 67
Jago Collins 11

If you are interested in Solutions to the Bongard Problems you may follow the previous link, but of course it is up to you to decide how much it is worth looking at those “spoilers” before spending some time to solve a problem.


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):

 
Bongard
 
Hofstadter
 
Foundalis
 
Insana
 
Shanahan
 
Howells
 
Gunnarsson
 
Gáspár
 
Lewis
 
Fairbanks
 
Collins    

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

 

Back to Harry’s introduction to his Ph.D. research page