Puzzles/Bongard problems/Solutions

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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 [1]. Thus, the table provided here is obsolete.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
Bottom figures slightly towards centerline Bottom figures slightly out from centerline
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