Operations depicted in right-sorted examples are called "commutative".

"Order matters" here means that if the objects in the top half were to switch places, the output would look different.

https://en.wikipedia.org/wiki/Commutative_property

Adjacent-numbered pages:
BP1152 BP1153 BP1154 BP1155 BP1156  *  BP1158 BP1159 BP1160 BP1161 BP1162

nice, abstract, unwordable, notso, structure, rules, miniworlds

Leo Crabbe

Jago originally designed this as a triptych. EX9655, EX9656, and EX9657 belong in a third category displayed further right of the two categories shown here. The third category is "all possible ways of matching are equally natural". - Aaron David Fairbanks, Apr 18 2022

Adjacent-numbered pages:
BP1186 BP1187 BP1188 BP1189 BP1190  *  BP1192 BP1193 BP1194 BP1195 BP1196

nice, abstract, creativeexamples, structure, miniworlds, dithering

Jago Collins

All examples in this Problem are paths of short and long line segments which do not self-intersect. This could be a useful world to use in an abstract binary sequence BP.

Adjacent-numbered pages:
BP1187 BP1188 BP1189 BP1190 BP1191  *  BP1193 BP1194 BP1195 BP1196 BP1197

nice, notso, arbitrary, traditional

[smaller | same | bigger]

Jago Collins

Adjacent-numbered pages:
BP1197 BP1198 BP1199 BP1200 BP1201  *  BP1203 BP1204 BP1205 BP1206 BP1207

nice, precise, allsorted, notso, color

Leo Crabbe

BP839 is about applying opposite transformations to a single object.

Adjacent-numbered pages:
BP1255 BP1256 BP1257 BP1258 BP1259  *  BP1261 BP1262 BP1263 BP1264 BP1265

easy, nice, abstract, arbitrary, anticomputer, left-null, structure, orderedtriplet, traditional, rules

Aaron David Fairbanks

All examples in this Bongard Problem are solid black shapes.

BP345 is "Intersection of circle and square vs. union of circle and square."

Adjacent-numbered pages:
BP1267 BP1268 BP1269 BP1270 BP1271  *  BP1273 BP1274 BP1275 BP1276 BP1277

EX1 is ambiguous because it might be a small circle minus a large square, or a small square minus a large circle.

easy, nice, precise, dual

Ben

Equivalently, some permutation of the objects reverses the sequence vs. not.

Palindromes fit left. Strings of distinct objects repeated any number of times fit left.

Adjacent-numbered pages:
BP1269 BP1270 BP1271 BP1272 BP1273  *  BP1275 BP1276 BP1277 BP1278 BP1279

nice, precise, allsorted, notso, sequence, miniworlds

[smaller | same | bigger]

Aaron David Fairbanks

A path is formally defined as a sequence of black squares such that no two squares are the same and every pair of consecutive black squares are orthogonally adjacent to each other.

Every grid on the left requires a minimum of an even number of paths such that every black square is a part of exactly one path, while every grid on the right requires a minimum of an odd number of paths to do so.

Adjacent-numbered pages:
BP1287 BP1288 BP1289 BP1290 BP1291  *  BP1293 BP1294

A grid with all whites fits on the left because it does not contain black squares, hence requiring 0 paths, an even number.

A grid with all blacks requires 1 path to go through all black squares (such as a zigzag through every row), an odd number.

The checkerboard grid on the left contains 12 black squares that cannot be joined with a path, hence it requires 12 paths, an even number.

nice, blackwhite, fixedgrid, left-listable, right-listable

Matt Nguyen