# 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 Thus, the table provided here is obsolete.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 |