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

Adjacent-numbered pages:
BP1268 BP1269 BP1270 BP1271 BP1272  *  BP1274 BP1275 BP1276 BP1277 BP1278

There are 6 ways of arranging the letters A, B and C: ABC, ACB, BAC, BCA, CAB, and CBA. The string "ABCABACBA" contains each of these as a substring, and would therefore be sorted left.

precise, allsorted, notso, sequence, traditional, miniworlds

[smaller | same | bigger]

Leo Crabbe

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

Sequences with a prime number of elements are sorted left when all their elements are unique, and sorted right otherwise.

Adjacent-numbered pages:
BP1270 BP1271 BP1272 BP1273 BP1274  *  BP1276 BP1277 BP1278 BP1279 BP1280

The sequence ABBABB would be sorted left, as it could be grouped into (AB)(BA)(BB), where each block is unique.

precise, unwordable, notso, sequence, miniworlds

Leo Crabbe

2D version of BP1275.

Adjacent-numbered pages:
BP1273 BP1274 BP1275 BP1276 BP1277  *  BP1279 BP1280 BP1281 BP1282 BP1283

precise, traditional, grid

Leo Crabbe

Rotation of the square is allowed.

See BP1280 for version with hexagons on a hexagonal grid.

Adjacent-numbered pages:
BP1274 BP1275 BP1276 BP1277 BP1278  *  BP1280 BP1281 BP1282 BP1283 BP1284

hard, precise, allsorted, unwordable, hardsort, left-finite, right-finite, left-full, fixedgrid, preciseworld

Leo Crabbe

See BP1279 for version with squares on a square grid.

Adjacent-numbered pages:
BP1275 BP1276 BP1277 BP1278 BP1279  *  BP1281 BP1282 BP1283 BP1284 BP1285

hard, precise, allsorted, unwordable, hardsort, left-finite, right-finite, preciseworld

Leo Crabbe

In particular, 40fps vs. 20fps.

Adjacent-numbered pages:
BP1281 BP1282 BP1283 BP1284 BP1285  *  BP1287 BP1288 BP1289 BP1290 BP1291

stub, precise, arbitrary, antihuman, animated, contributepairs, right-couldbe, preciseworld

Leo Crabbe

Solution worded differently:

Colors that use the primary color red to make, vs colors that don’t/cannot

Reds and its secondary and tertiary relations vs. not

Red and its pigment relatives vs. not

Comment from the author (a.k.a. Morgan Kidd):

Thank you Sally D. for introducing me to bongards!

Adjacent-numbered pages:
BP1285 BP1286 BP1287 BP1288 BP1289  *  BP1291 BP1292 BP1293 BP1294

precise, notso, color

Morgan

Left hand examples can be thought of as "graph circles".

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

Adjacent-numbered pages:
BP1286 BP1287 BP1288 BP1289 BP1290  *  BP1292 BP1293 BP1294

precise, unwordable, notso, left-narrow

Leo Crabbe

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

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