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BP1273 Sequence contains each possible way its distinct elements can be arranged as a subsequence vs. not so.
(edit; present; nest [left/right]; search; history)
REFERENCE

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

CROSSREFS

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

EXAMPLE

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.

KEYWORD

precise, allsorted, notso, sequence, traditional, miniworlds

CONCEPT sequence (info | search),
overlap (info | search)

WORLD

[smaller | same | bigger]

AUTHOR

Leo Crabbe

BP1274 Reversing the sequence permutes the objects vs. not.
(edit; present; nest [left/right]; search; history)
COMMENTS

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.

CROSSREFS

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

KEYWORD

nice, precise, allsorted, notso, sequence, miniworlds

WORLD

[smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP1275 There is a way of grouping elements into (more than one) equal-sized blocks such that no block appears twice vs. there exists no such grouping.
(edit; present; nest [left/right]; search; history)
COMMENTS

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

CROSSREFS

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

EXAMPLE

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

KEYWORD

precise, unwordable, notso, sequence, miniworlds

CONCEPT element_grouping (info | search)

AUTHOR

Leo Crabbe

BP1276 Ways of representing the sequence "ABABCBACCBAC" by grouping its elements into equal-sized blocks and relabelling them (identical blocks are represented by the same element) vs. representations of different sequences.
(edit; present; nest [left/right]; search; history)
COMMENTS

The solver isn't expected to word their solution as it is put above, but to notice that the sequences on the left all "agree" with each other under the aforementioned grouping rule. The 12-element sequence in EX10249 can be "squashed down" into the 6-element sequence in EX10251, which in turn can be "squashed down" into the 3-element sequence in EX10252.

CROSSREFS

Compare to BP121

Adjacent-numbered pages:
BP1271 BP1272 BP1273 BP1274 BP1275  *  BP1277 BP1278 BP1279 BP1280 BP1281

KEYWORD

unwordable, notso, arbitrary, sequence

CONCEPT element_grouping (info | search)

AUTHOR

Leo Crabbe

BP1282 If two players take turns moving moving the black circles with the intention of capturing their opponent's piece, one can always "checkmate" the other vs. the game results in a draw if the players play optimally.
(edit; present; nest [left/right]; search; history)
COMMENTS

Players can only move their piece to a node connected to their current position. A win is secured by moving to a node your opponent is occupying.


Which player that can force a win in left-sorted examples can change depending on who moves first.

CROSSREFS

See BP1284 for an animated Problem about the same game.

Adjacent-numbered pages:
BP1277 BP1278 BP1279 BP1280 BP1281  *  BP1283 BP1284 BP1285 BP1286 BP1287

KEYWORD

allsorted, unwordable, notso, teach

CONCEPT capture_game (info | search)

AUTHOR

Leo Crabbe

BP1290 Red and its derivative hues vs. not
(edit; present; nest [left/right]; search; history)
COMMENTS

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!

CROSSREFS

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

KEYWORD

precise, notso, color

AUTHOR

Morgan

BP1291 Black points are the set of vertices that are some shortest-distance away from some white vertex vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

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

CROSSREFS

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

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

KEYWORD

precise, unwordable, notso, left-narrow

CONCEPT graph (info | search)

AUTHOR

Leo Crabbe

BP1293 Line segment separates the dots into two consecutive Fibonacci numbers vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

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.

REFERENCE

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

CROSSREFS

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

EXAMPLE

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.

KEYWORD

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

CONCEPT addition (info | search),
number (info | search),
dot (info | search),
sides_of_line (info | search),
sequence (info | search)

AUTHOR

Matt Hodges

BP1294 No sequence starts with another vs. some sequence starts with another.
(edit; present; nest [left/right]; search; history)
REFERENCE

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

CROSSREFS

Adjacent-numbered pages:
BP1289 BP1290 BP1291 BP1292 BP1293  * 

KEYWORD

allsorted, notso, preciseworld, dithering, collection

AUTHOR

Leo Crabbe

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