See BP334 for a version of the same idea, but using arbitrary shapes instead of dots.

Adjacent-numbered pages:
BP329 BP330 BP331 BP332 BP333  *  BP335 BP336 BP337 BP338 BP339

precise, allsorted, number, math, left-narrow, right-narrow, right-null, help, traditional, preciseworld

dots [smaller | same | bigger]

Aaron David Fairbanks

All examples in this Problem are position-independent Bongard Problems.

Positioning here includes objects' positions within the panels and objects' positions relative to each other.

There are very subtle distinctions to be made between the usage of variance of position in these BPs for the sake of noise (obscuring the solution eg. BP557), clarity (generalising the solution to make it more fundamental eg. BP79) or help (aiding the observer in finding the solution eg. BP334). There is certainly a degree of overlap between these three definitions, they are not disconnected.

Adjacent-numbered pages:
BP794 BP795 BP796 BP797 BP798  *  BP800 BP801 BP802 BP803 BP804

meta (see left/right), links

Leo Crabbe

See BP334 for a version of the same idea, but using dots instead of arbitrary shapes.

Adjacent-numbered pages:
BP197 BP198 BP199 BP200 BP201  *  BP203 BP204 BP205 BP206 BP207

noisy, number, traditional

Giuseppe Insana

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