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

Adjacent-numbered pages:
BP1278 BP1279 BP1280 BP1281 BP1282  *  BP1284 BP1285 BP1286 BP1287 BP1288

Leo Crabbe

In the depicted "capture game", the objective is to capture your opponent's piece by moving to a node they are occupying. Players take turns moving their pieces. You can only move to a node that is linked to yours. Optimal play can either lead to a win-lose state or a draw state.

A frame where only one black disc is visible signifies that a player has taken the other's piece, winning the game.

Adjacent-numbered pages:
BP1279 BP1280 BP1281 BP1282 BP1283  *  BP1285 BP1286 BP1287 BP1288 BP1289

teach, animated

Leo Crabbe

In the depicted "capture game", the objective is to capture your opponent's piece by moving to a node they are occupying. Players take turns moving their pieces. You can only move to a node that is linked to yours. All examples are "boards" where either player can force a win, depending on who moves first.

Players are assumed to be making optimal choices.

Adjacent-numbered pages:
BP1280 BP1281 BP1282 BP1283 BP1284  *  BP1286 BP1287 BP1288 BP1289 BP1290

stub

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

Adjacent-numbered pages:
BP1282 BP1283 BP1284 BP1285 BP1286  *  BP1288 BP1289 BP1290 BP1291 BP1292

meta (see left/right), metaconcept

bp [smaller | same | bigger]

Leo Crabbe

Adjacent-numbered pages:
BP1283 BP1284 BP1285 BP1286 BP1287  *  BP1289 BP1290 BP1291 BP1292 BP1293

world, animated

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

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

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