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
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 metaBongard
Problems (and metametaBP’s) that the author and Joseph Insana, in
cooperation, came up with. What is meant by the term “metaBP” can be understood
by looking at problem #200, above. The best description of metaBP’s on the web
can be found under
these pages by Joseph Insana.
Implemented metaBP’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.
Colorcoding showing which concept was originally introduced by which
designer (listed chronologically):

Bongard 

Hofstadter 

Foundalis 

Insana 

Shanahan 

Howells 

Gunnarsson 

Gáspár 

Lewis 

Fairbanks 

Collins 


Notes:

Those among the concepts listed below that
are primitive are marked with this symbol:
on the leftmost column. A composite (nonprimitive) 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.

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.

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 alreadyknown 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”.

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 2D, blackandwhite, visual cognition? Since the list has been
constructed by many people contributing to it, it can be viewed as the
endproduct 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 



Chesslike 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 (Xpoints) 
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 



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 3D) 
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 2D 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 (Xpoints) 

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 

Metasolution 
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 



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 


Selfreference 
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 



Speed (or velocity) 
300,
358 

Square (as opposed to rectangle) 
242 

Square of a number 
384 

Sum (arithmetic) 
see: Addition (arithmetic) 




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
(Tpoints) 
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 



True, correct / False, incorrect 
393 

Trunk of a treelike 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: 3D Depth 

Visual illusion 
237 

Wave 
381 

Wrapping around the box 
244 

Written in
singlestroke 
271,
340 

3D Depth 
195 

3D Impossible solid /
Possible solid 
252 

3D Net 
371,
385 

3D Solid 
250,
251, 370,
371 
