Right examples are called "derangements".

Adjacent-numbered pages:
BP926 BP927 BP928 BP929 BP930  *  BP932 BP933 BP934 BP935 BP936

handed, leftright, sequence, traditional, left-listable, right-listable

dot_clusters_sequence_horizontal [smaller | same | bigger]

Aaron David Fairbanks

Attributes are shading, shape, and number.

There are always three groups.

This problem is related to the card game Set.

Adjacent-numbered pages:
BP1133 BP1134 BP1135 BP1136 BP1137  *  BP1139 BP1140 BP1141 BP1142 BP1143

nice, precise, allsorted, notso, preciseworld

William B Holland

The triangles vary in size in order to make the solution clearly about quantity, not "total mass".

Adjacent-numbered pages:
BP1203 BP1204 BP1205 BP1206 BP1207  *  BP1209 BP1210 BP1211 BP1212 BP1213

stub, precise, spectrum, dual, handed, leftright, traditional

Aaron David Fairbanks

Based on BP91, which is the same, except for that the right side is "four identical elements".

Adjacent-numbered pages:
BP1221 BP1222 BP1223 BP1224 BP1225  *  BP1227 BP1228 BP1229 BP1230 BP1231

stub, arbitrary, number, traditional

Aaron David Fairbanks

All examples show a single straight line segment and some dots, with no dot lying on the line. In left examples the line depicts the Fibonacci recurrence: the two groups of dots are consecutive Fibonacci numbers, and their total is the next Fibonacci number. Here 0 counts as a Fibonacci number (0, 1, 1, 2, 3, 5, ...), so a lone dot with the line beside it fits left as the split 0|1. The six original left examples give each of the splits 0|1, 1|1, 1|2, 2|3, 3|5, and 5|8 exactly once, so their totals run through 1, 2, 3, 5, 8, 13. Several right examples are near misses: 2|5 uses two Fibonacci numbers that are not adjacent in the sequence, 3|3 repeats a term instead of pairing neighbors, and 0|3 has a Fibonacci total but the wrong split. A dot lying exactly on the dividing line would be ambiguous, so such dots are excluded. This problem was created by hand as a test for AI models that could not have previously seen it in training; see the reference for details.

M. Hodges, Bongard Problems, matthodges.com, 19 Aug 2026. https://matthodges.com/posts/2026-08-19-bongard-problems/

Compare BP334 (even number of dots vs. odd number of dots) and BP202 (even number of shapes vs. odd number of shapes), other Problems solved by counting rather than by geometry.

Adjacent-numbered pages:
BP1288 BP1289 BP1290 BP1291 BP1292  *  BP1294 BP1295

The box with five dots on one side of the line and eight on the other fits on the left because 5 and 8 are consecutive Fibonacci numbers, with 13, their total, the next one. The box with two dots on one side and five on the other fits on the right because 2 and 5, although both Fibonacci numbers, are not consecutive. The box with three dots all on one side fits on the right because its split is 0|3 rather than the 1|2 a Fibonacci total of 3 would need.

precise, notso, number, math, left-narrow, traditional

Matt Hodges