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 Thus, the table provided here is obsolete.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 Hofstadter150 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) FoundalisBottom 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