Puzzles/Bongard problems/Solutions

Please note that as of September 26, 2019, Harry Foundalis has added new problems to his index and changed a number of old ones. He has also supplied official solutions at . Thus, the table provided here is obsolete.

The table shows what is differentiated from the left and right.

The problems themselves are available at www.foundalis.com/res/bps/bpidx.htm.

#	Left	Right	Creator
1	Empty	Not empty	Bongard
2	Big figures	Small figures	Bongard
3	Outlined figures	Filled figures	Bongard
4	Convex figures	Concave figures	Bongard
5	Boxy shapes	Curvy shapes	Bongard
6	Triangles	Quadrilaterals	Bongard
7	Stretched vertically	Stretched horizontally	Bongard
8	Right side	Left side	Bongard
9	Smooth contours	Pointy contours	Bongard
10	Triangles	Quadrilaterals	Bongard
11	Elongated shapes	Not elongated	Bongard
12	Elongated convex hull	Not elongated convex hull	Bongard
13	Vertical rectangles, horizontal ellipses	Horizontal rectangles, vertical ellipses	Bongard
14	Large total line length	Small total line length	Bongard
15	Closed regions	Open regions	Bongard
16	Counterclockwise spiral	Clockwise spiral	Bongard
17	Angle directed inward	No inward angle	Bongard
18	Neck	No neck	Bongard
19	Horizontal neck	Vertical neck	Bongard
20	Dots on same side of neck	Dots on different sides	Bongard
21	Small figure present	No small figures present	Bongard
22	Sizes approximately equal	Sizes differ	Bongard
23	One figure	Two figures	Bongard
24	Circle present	No circles	Bongard
25	Triangles are filled	Circles are filled	Bongard
26	Filled triangle	No filled triangle	Bongard
27	More shaded figures	More outlined figures	Bongard
28	More filled circles	More outlined circles	Bongard
29	More circles inside	More circles outside	Bongard
30	Line crosses itself	Line does not cross by itself	Bongard
31	One line	Two lines	Bongard
32	Sharp projection	No sharp projection	Bongard
33	Acute angle	No acute angle	Bongard
34	Big hole	Small hole	Bongard
35	Hole parallel to figure	Hole perpendicular to figure	Bongard
36	Triangle above circle	Circle above triangle	Bongard
37	Triangle above circle	Circle above triangle	Bongard
38	Triangle larger than circle	Circle larger than triangle	Bongard
39	Lines almost parallel	Not all lines parallel	Bongard
40	Three collinear points	No collinear points	Bongard
41	White points in a line	White points make a triangle	Bongard
42	Enclosed points in a line	Enclosed points make a triangle	Bongard
43	Squiggle bigger on right	Squiggle bigger on left	Bongard
44	Both branches have a dot	One branch has both dots	Bongard
45	White shape covers part of black	Black shape covers part of white	Bongard
46	Triangle covers part of circle	Circle covers part of triangle	Bongard
47	Triangle in circle	Circle in triangle	Bongard
48	Black above white	White above black	Bongard
49	Enclosed dots close, outer dots spread out	Enclosed dots spread out, outer dots close	Bongard
50	Figure is (approximately) left-right symmetric	Figure is not symmetric	Bongard
51	Two dots are close together	No two dots are close	Bongard
52	Arrows point opposite ways on track	Arrows point in same way on track	Bongard
53	Outer polygon has more sides	Inner polygon has more sides	Bongard
54	Plus-Circle-Triangle goes clockwise	Plus-Circle-Triangle goes counter-clockwise	Bongard
55	Dot is a bit counter-clockwise of notch	Dot is a bit clockwise of notch	Bongard
56	Colors match	Both black and white shapes	Bongard
57	Two copies of an object	Two different objects	Bongard
58	Black squares are same size	Black squares are different size	Bongard
59	Same shape, different size (similar objects)	Not similar objects	Bongard
60	Two shapes are similar	No two shapes are similar	Bongard
61	Line splits plusses into equal numbers	Line splits plusses into unequal numbers	Bongard
62	The two ends of the line are far apart	The two ends of the line are close together	Bongard
63	Shadow at right side	Shadow at left side	Bongard
64	Small circle is not on axis of ellipse	Small circle is on axis of ellipse	Bongard
65	Triangles are on a horizontal line	Triangles are arranged vertically	Bongard
66	Unconnected dots are on a horizontal line	Unconnected dots are on a vertical line	Bongard
67	Right branch attached above left branch	Left branch attached above right branch	Bongard
68	Right branch goes higher than left branch	Left branch goes higher than right branch	Bongard
69	Dot is on central stem	Dot is on a side branch	Bongard
70	All branches come from central stem	Branches have branches	Bongard
71	Three levels of nesting	Two levels of nesting	Bongard
72	The line ends are parallel to each other	The line ends are perpendicular to each other	Bongard
73	Ellipse and rectangle are perpendicular	Ellipse and rectangle are parallel	Bongard
74	Body is widest at base	Body is widest at top	Bongard
75	Triangle is located at hollow side of arc (inside)	Triangle is located "outside" the arc	Bongard
76	The long sides are curved inward	Long sides are curved outward	Bongard
77	(Approximately) equal angles	Unequal angles	Bongard
78	Three lines (extended) meet at a point	Three lines not concurrent	Bongard
79	Black dot closer to circle than to triangle	Black dot closer to triangle than to circle	Bongard
80	Plus is equidistant from dots	Plus is closer to one dot	Bongard
81	Black and white objects can be separated by a line	No line separates white from black	Bongard
82	Three of the pluses always form an (approximately) equilateral triangle	No three of the pluses form equilateral triangle	Bongard
83	Circle inside plus region	Circle outside plus region	Bongard
84	Square outside "necklace"	Square inside "necklace"	Bongard
85	Three lines	Five lines	Bongard
86	Three branches	Five branches	Bongard
87	Four segments	Five segments	Bongard
88	Three pills	Five pills	Bongard
89	Three groups	Five groups	Bongard
90	Three white groups	Four white groups	Bongard
91	Three of something	Four of something	Bongard
92	Single chain with two ends	Branched chain	Bongard
93	Branch point is white	Branch point is black	Bongard
94	Black in middle	Black at end	Bongard
95	Vertical stripes	Horizontal stripes	Bongard
96	Triangle	Quadrilateral	Bongard
97	Triangular form	Circular form	Bongard
98	Triangle	Quadrilateral	Bongard
99	Circle necklace crosses triangle necklace	Circle necklace doesn't cross triangle necklace	Bongard
100	Letter А	Letter Б	Bongard
101	Two horizontal indentations	One horizontal and one vertical indentation	Hofstadter
102	Adornments on track point out	Adornments point out-in-in-out	Hofstadter
103	Line plus point forms isosceles triangle	Line plus point forms scalene triangle	Hofstadter
104	One circle passes through center of other	Circle centers disjoint from circles	Hofstadter
105	Ends of connecting line point the same way	Line ends do not point the same way	Hofstadter
106	Line through dots has negative slope	Line through dots has positive slope	Hofstadter
107	Three plain sides	Three not plain sides	Hofstadter
108	Protrusions smaller at ends	Protrusions larger at ends	Hofstadter
109	Circle to the right of average plus position	Circle to the left of average plus position	Hofstadter
110	Four circles	More or less than four circles	Hofstadter
111	Intermediate shape is triangle	Intermediate shape is square or circle	Hofstadter
112	Evenly spaced horizontally	Evenly spaced vertically	Hofstadter
113	"T" junction	"Y" junction	Hofstadter
114	Four crossings	Two crossings	Hofstadter
115	Outermost and innermost shapes are connected	Outermost and innermost shapes not connected	Hofstadter
116	Bottom of polygon is a side	Bottom of polygon is a point	Hofstadter
117	Triangle points to center of circle	Triangle does not point to center of circle	Hofstadter
118	No cycles	Has a cycle	Hofstadter
119	One circle with small deviations	Something else	Hofstadter
120	Only right turns or only left turns	Mixture of left and right turns	Hofstadter
121	Pairs on top determine bottom: ∨∨/O ∨∧/☐ ∧∨/ ∧∧/▽	Top is flipped: ∧∧/O ∧∨/☐ ∨∧/ ∨∨/▽	Hofstadter
122	Line divides interior into two regions	Something else	Hofstadter
123	Two shapes with similar silhouettes	Two shapes with dissimilar silhouettes	Hofstadter
124	Shapes have same colour	Shapes have different colour	Hofstadter
125	Indentation and protrusion have same shape	Two indentations or differing shapes	Hofstadter
126	One big and one small shape	Two shapes of the same size, or three shapes	Hofstadter
127	One circle	An even number of circles	Hofstadter
128	Inside shapes match outside shapes	Inside shapes differ from outside shapes	Hofstadter
129	Circular indentation is on a protrusion	Circular indentation is on an indentation	Hofstadter
130	Big shape has curves	Big shape has no curves	Hofstadter
131	Four dots form a parallelogram	Four dots do not form a parallelogram	Hofstadter
132	One dot above other two	One dot below other two	Hofstadter
133	Circle centre collinear with dots	Circle centre not collinear with dots	Hofstadter
134	Circle centres are collinear	Circle centres not collinear	Hofstadter
135	Circles on same side of squiggle (can meet by rolling)	Squiggle separates circles	Hofstadter
136	Polygon is concave	Polygon is convex	Hofstadter
137	Number of dots matches number of polygon sides	Numbers don't match	Hofstadter
138	All dots are on the perimeter of their convex hull	One dot is inside the convex hull	Hofstadter
139	Sequence follows simple pattern	Sequence is irregular	Hofstadter
140	Two small things match, large thing differs	Something else	Hofstadter
141	Three and two	Three and one and one	Hofstadter
142	Three and two, two are adjacent in sequence	The two are not sequentially adjacent	Hofstadter
143	Three and two	Four and one	Hofstadter
144	Different features yield different 3-2 splits	Different features yield same 3-2 split	Hofstadter
145	Different features yield same 4-1 split	Different features yield different 4-1 splits	Hofstadter
146	A shape contains another if and only if it is a circle	A shape contains another if and only if it is a triangle	Hofstadter
147	Three in a line and two vertical	Three in a line and two horizontal	Hofstadter
148	A little bit missing from polygonal figure	A little bit extra on polygonal figure	Hofstadter
149	One square is alone	One circle is alone	Hofstadter
150	Even number of sides	Odd number of sides	Hofstadter
	Odd number of squares	Even number of squares
	Two matching shapes and a square (the matching shapes can be squares)	Two matching shapes and either a circle or a triangle
151	Ignoring the circle closest to the cross, the remaining circles form an equilateral triangle	Ignoring the circle closest to the cross, the remaining circles do not form an equilateral triangle	Hofstadter
152	Not left-right symmetric	Left-right symmetrical	Hofstadter
153	No "staples"	Has "staple" shapes by some line ends	Hofstadter
154	45 degree flat ends	Rounded ends	Hofstadter
155	Curly ends	Straight ends	Hofstadter
156	Three groups	Two groups	Hofstadter
157	White background	Black background	Foundalis
158	Roughly 30 degree slope	Roughly 15 degree slope	Foundalis
159	Rectangle	Triangle	Foundalis
160	Quadrilateral is almost a triangle	Quadrilateral is not almost a triangle	Foundalis
161	Parallel lines // Collinear midpoints	Lines not parallel // Midpoints not collinear	Foundalis
162	Extending every other line segment makes them converge at a point	Not the case	Foundalis
163	Identical shapes can be joined by a straight line that is not intersected by the third shape	Identical shapes cannot be joined by an unintersected straight line	Foundalis
164	# of shapes is number of sides on largest - 1	# of shapes is number of sides on largest + 1	Foundalis
165	Segment roughly perpendicular to a polygon side	Segment parallel to a polygon side	Foundalis
166	Two groups	Three groups	Foundalis
167	Clusters come in pairs	Clusters come in triples	Foundalis
168	Bulky interior	Narrow Interior	Foundalis
169	Some shape appears once	Shapes appear an even number of times	Foundalis
170	Overall "arch" shape	Overall "X" shape	Foundalis
171	Angle bisectors meet at the incentre	Perpendicular bisectors meet at orthocentre	Foundalis
172	Has rotational symmetry	No rotational symmetry	Foundalis
173	All segments within about 30 degrees of each other	Segments at all angles	Foundalis
174	Middle shape is convex	Middle shape is not convex	Foundalis
175	Small shape can slide into "C"	Shape on right cannot slide into "C"	Foundalis
176	Short (almost straight) path between dots	Long (winding) path between dots	Foundalis
177	All regions are convex	There exist concave regions	Foundalis
178	Circle centred on perpendicular triangle	No perpendicular triangle, or circle not centred on it	Foundalis
179	Bigger at bottom	Bigger at top	Foundalis
180	White side of boundary getting wider	Black side of boundary getting wider	Foundalis
181	One indentation	Two indentations	Foundalis
182	Overall shape is concave	Overall shape is convex	Foundalis
183	Ornamental endings on same side of central line	Ornamental endings on opposite sides	Foundalis
184	Central part of line is smooth	Central part of line has a zig-zag	Foundalis
185	Ornamental regions at both ends	Ornamental regions at both ends and at middle	Foundalis
186	Repetition of a simple theme	Repetition of a multi-part theme	Foundalis
187	Inner polygons have one more side than overall shape	Inner polygons have one side fewer than overall shape	Foundalis
188	Triangles made of squares, squares made of triangles	Triangles made of triangles, squares made of squares	Foundalis
189	Each group is all outlines or all solids	Some group has both a solid and an outline	Foundalis
190	Items on a line have the same color	Some line has items of both colors	Foundalis
191	Entrance is on left	Entrance is on right	Foundalis
192	Elongated vertically	Elongated horizontally	Foundalis
193	White triangle	White rectangle	Foundalis
194	Outer shape is rectangle	Outer shape is triangle	Foundalis
195	Crossing your eyes to perceive the two groups of objects separated by a line as one group gives the effect of the lower objects being further away	The effect is reversed (the lower objects look closer)	Foundalis
195	Bottom figures slightly towards centerline	Bottom figures slightly out from centerline	Foundalis
196	Lighter	Darker	Foundalis
197	Different font (letters more connected)	Different font (letters less connected)	Foundalis
198	One end of chain is inside large shape	Both ends of chain are outside large shape	Foundalis
199	Stable if placed on ground	Would fall over if placed on ground	Foundalis
200	Differences not related to number	Distinction depends on number of something	Foundalis
201	One of the shapes can imbricate into another one	No matching pair	Insana
202	Even number of shapes	Odd number of shapes	Insana
203	Number of shapes is a prime number	Number of shapes not prime	Insana
204	Picture composed of 5 shapes or lines	Picture composed of 4 shapes or lines	Insana
205	Similar shapes come in pairs	At least one type of shape has odd number	Insana
206	End of spiral is vertical	End of spiral is horizontal	Insana
207	Ends of the spirals are on the left half of the panel	Ends of the spirals are on the right half of the panel	Insana
208	Spirals all leftward	Spirals all rightward	Insana
209	There are no objects centered in the box	There is an object centered in the box	Insana
210	All black regions (not dots) touch the edge of the frame	At least one black region (not dots) does not touch the edge of the frame	Insana
211	More black than white	More white than black	Insana
212	Square dots have small spacing	Dots spacing is larger	Insana
213	Inner extremity of the spiral crossses the line	There is no crossing at the inner extremity of spiral	Insana
214	Picture makes a smiling face	Various other pictures	Insana
215	The longest curve is descending and then ascending (picture makes a happy face)	The longest curve is ascending and then descending (picture makes unhappy face)	Insana
216	Line ends with a wider dot	Line has constant thickness	Insana
217	Shapes are fully coloured	Shapes have small white spaces in them	Insana
218	Incomplete patterns, sequences, or shapes	"Overcomplete" patterns, sequences, or shapes	Insana
219	The lines are parallel	The lines are not parallel	Insana
220	One cluster (or shape)	Two clusters (or shapes)	Insana
221	No rectangles at the ends of the line	Rectangles at the ends of the line	Insana
222	No complete outline of a rectangle	Complete outline of a rectangle	Insana
223	Big shape is a disk	Big shape is a slightly flattened disk	Insana
224	The figures have vague central symmetry	The figures do not have any symmetry	Insana
225	Low contrast	High contrast	Insana
226	Protrusions all bend clockwise	No protrusions or protrusions do not all bend clockwise	Insana
227	The black shapes touch the edge of the frame if and only if the black dots do not touch each other	The black shapes touch the edge if and only if the dots touch each other	Insana
228	All figures are derived from a “J” shape	All figures are derived from a “L” shape	Insana
229	It is possible to rotate one of the two figures and place it next to the other figure so that the dotted edges of the figures touch; the combined figures form some number of three by one rectangles. Each rectangle is wholly black or white	The figures either do not form three by one rectangles, or the formed rectangles are not wholly black or white	Insana
230	There are seven "outer" vertices of the shapes (the Xs have four vertices)	There are another number of vertices	Insana
231	The total number of internal angles larger than 180 degrees is 5	The total number of internal angles larger than 180 degrees is not 5	Insana
232	There are six white regions	There are not six white regions	Insana
233	Half of the shapes are stars	Half of the shapes are lines	Shanahan
234	Circle, if dropped, winds up on the right	Circle, if dropped, winds up on the left	Shanahan
235	The "tail" of the circle is long enough to reach the other circle	The "tail" cannot reach the other circle	Shanahan
236	Darker spots	No darker spots	Shanahan
237	Bottom central line looks longer (note Müller-Lyer illusion)	Top central line looks longer (note Müller-Lyer illusion)	Shanahan
238	C's face the same direction	C's face opposite directions	Foundalis
239	Segment fits curve opening without rotation	Segment fits curve opening if rotated	Foundalis
240	Ends have line of sight	Ends have no line of sight	Howells
241	Circle circumference less than line length	Circle circumference greater than line length	Howells
242	The dots all lie on the edges of a square with a dot at each corner	The dots don't all lie on the edges of a square with a dot at each corner	Howells
243	The shape closest to a corner is the triangle	The shape closest to a corner is a circle	Howells
244	Scanning row by row, left to right, the number of white squares between the black squares is constant	The number of white squares between the black squares varies	Howells
245	Triangle attached to edge of tetrahedron	Any other graph	Gunnarsson
246	Each triangle (if any) points to exaclty one circle (triangles cannot point at circles they are inside) and each circle is pointed to by at most one triangle	Either not all triangles point to a circle or a circle has two triangles pointing at it	Gunnarsson
247	Five curved or straight line segments	Any other number of curved or straight line segments	Gunnarsson
248	The dashed outline is a slightly shrunk copy of the solid outlined shape. The dashed outline is not translated or rotated	Two shapes (dashed or solid-line) arranged otherwise	Gunnarsson
249	The dashed outline is a shrunk copy of the solid outlined shape. The dashed outline is not translated but can be rotated as long as it remains inside the larger shape	Two shapes (dashed or solid-line) arranged otherwise	Gunnarsson
250	Tetrahedron	Not a tetrahedron	Gunnarsson
251	Regular tetrahedron (approximately)	Not a regular tetrahedron	Gunnarsson
252	Impossible objects (not projections of actual three dimensional solids)	Projections of three dimensional solids	Gunnarsson
253	Prime total number of objects	Composite total number of objects	Gunnarsson
254	There exists a pair of vertices that the circle is collinear with	There does not exist a pair of vertices that the circle is collinear with	Gunnarsson
255	There is one and only one pairing resulting in three pairs of parallel lines	There is not one and only one pairing resulting in three pairs of parallel lines	Gunnarsson
256	Every circle's closest neighbour is a different size	not every circle's closest neighbour is a different size	Gunnarsson
257	The distance between the dot and the right edge of the panel is equal to the distance between the middle of the line and the bottom edge of the panel	The distance between the dot and the right edge of the panel is not equal to the distance between the middle of the line and the bottom edge of the panel	Gunnarsson
258	Shapes are reflections of each other over the x-axis	Shapes are reflections of each other over the y-axis	Gunnarsson
259	Front shape is same as back shape rotated slightly clockwise and partially obscures bottom shape	Two shapes arranged otherwise	Gunnarsson
260	There are two shapes that lie on the outline of the larger shape and they are smaller versions of the larger shape and each are rotated by 90 degrees	There are not two shapes that lie on the outline of the larger shape or they are not smaller versions of the larger shape or they are not 90 degree rotations of the larger shape	Gunnarsson
261	Line between black dots crosses line between white dots	Line between black dots doesn't cross line between white dots	Ihde
262	Odd number of dots in sharp bends of line	Even number of dots in sharp bends of line	Barenbaum
263	One straight end line segment	No straight end line segment	Barenbaum
264	Letter has enclosed region	No enclosed region	Gáspár
265	Letter has line of symmetry	Does not have line of symmetry	Gáspár
266	Text contains junction (point receiving 3 lines).	Text does not contain junction	Gáspár
267	Odd number of lines	Even number of lines	Gáspár
268	Symbol's profile is taller than it is wide	Symbol's profile wider than it is tall	Gáspár
269	Rotational symmetry	No rotational symmetry	Gáspár
270	Lines vertical or horizontal	None vertical or horizontal	Gáspár
271	Can draw in one stroke without retracing (intersections permitted)	Requires retracing or more than one stroke	Gáspár
272	Three end points (triangular)	Not three end points (rectangular)	Gáspár
273	Stable	Unstable	Gáspár
274	Symbols would collect water if it rained	Symbols would not collect water if it rained	Gáspár
275	The two end points of the symbol have line of sight	The two end points of the symbol do not have line of sight	Gáspár
276	There are line segments not part of loops	Symbols are made up of only loops	Gáspár
277	One connected shape	Two or more disconnected shapes	Gáspár
278	Symbol on the left is shorter	Symbol on the right is shorter	Gáspár
279	Symbol on the left has greater black area	Symbol on the right has greater black area	Gáspár
280	Symbol on the left is wider	Symbol on the right is wider	Gáspár

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