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BP569 Triangular number of dots vs. non-triangular number of dots
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COMMENTS

All examples in this Problem are groups of black dots.


The nth triangular number is the sum over the natural numbers from 1 to n, where n > 0. Note: 0 is the 0th triangular number. The first few triangular numbers are 0, 1, 3 (= 1+2) and 6 (= 1+2+3)

CROSSREFS

Adjacent-numbered pages:
BP564 BP565 BP566 BP567 BP568  *  BP570 BP571 BP572 BP573 BP574

KEYWORD

nice, precise, allsorted, notso, number, math, left-narrow, left-null, help, preciseworld

WORLD

dots [smaller | same | bigger]

AUTHOR

Leo Crabbe

BP564 Discrete points intersecting boundary of convex hull vs. connected segment intersecting boundary of convex hull
(edit; present; nest [left/right]; search; history)
COMMENTS

If a "string" is wound tightly around the shape, does one of its segments lie directly on the shape?


All examples in this Problem are connected line segments or curves.


We are taking lines here to be infinitely thin, so that if the boundary of the convex hull intersects the endpoint of a line exactly it is understood that they meet at 1 point.

CROSSREFS

Adjacent-numbered pages:
BP559 BP560 BP561 BP562 BP563  *  BP565 BP566 BP567 BP568 BP569

EXAMPLE

Imagine wrapping a string around the pointed star. This string would take the shape of the boundary of the star's convex hull (a regular pentagon), and would only touch the star at the end of each of its 5 individual tips, therefore the star belongs on the left.

KEYWORD

hard, nice, allsorted, solved, perfect

AUTHOR

Leo Crabbe

BP562 There exists a closed trail that hits each vertex exactly once vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

Left examples are called "Hamiltonian graphs."

CROSSREFS

Adjacent-numbered pages:
BP557 BP558 BP559 BP560 BP561  *  BP563 BP564 BP565 BP566 BP567

KEYWORD

math, traditional

CONCEPT graph (info | search),
distinguishing_crossing_curves (info | search)

WORLD

connected_graph [smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP560 There exists a closed trail that hits each edge exactly once vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

Left examples are called "Eulerian graphs."


A connected graph is Eulerian if and only if each vertex is incident to an even number of edges.

CROSSREFS

Adjacent-numbered pages:
BP555 BP556 BP557 BP558 BP559  *  BP561 BP562 BP563 BP564 BP565

KEYWORD

precise, allsorted, math, traditional, preciseworld

CONCEPT graph (info | search),
distinguishing_crossing_curves (info | search),
all (info | search),
even_odd (info | search),
existence (info | search)

WORLD

connected_graph [smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

BP559 Cross section of a cube vs. not cross section of a cube
?
(edit; present; nest [left/right]; search; history)
COMMENTS

All examples are solid black shapes.


This problem is absurdly hard. It makes a good extreme example. - Aaron David Fairbanks, Nov 23 2020

CROSSREFS

Adjacent-numbered pages:
BP554 BP555 BP556 BP557 BP558  *  BP560 BP561 BP562 BP563 BP564

KEYWORD

hard, precise, allsorted, notso, stretch, challenge, left-narrow, perfect

CONCEPT cube (info | search),
cross_section (info | search)

WORLD

fill_shape [smaller | same | bigger]

AUTHOR

Leo Crabbe

BP558 Point sequence that is increasing or decreasing in height vs. point sequence that alternates in height
(edit; present; nest [left/right]; search; history)
COMMENTS

All examples in this Problem are groups of 3 dots.


Any example where 2 adjacent dots have the same height would be ambiguously sorted.

CROSSREFS

Adjacent-numbered pages:
BP553 BP554 BP555 BP556 BP557  *  BP559 BP560 BP561 BP562 BP563

EXAMPLE

Reading from right to left in the first box on the left hand side: the 2nd dot is higher than the 1st, and the 3rd is higher than the 2nd, so the sequence of dots is strictly increasing in height.

KEYWORD

nice, precise, antihuman, orderedtriplet, preciseworld

CONCEPT coordinate (info | search)

WORLD

three_points [smaller | same | bigger]

AUTHOR

Leo Crabbe

BP557 Equal horizontal length vs. not
(edit; present; nest [left/right]; search; history)
COMMENTS

All examples in this Problem are pairs of straight line segments.


This problem communicates the idea of projected distance, in this case from 2D to 1D (x-axis).

CROSSREFS

Adjacent-numbered pages:
BP552 BP553 BP554 BP555 BP556  *  BP558 BP559 BP560 BP561 BP562

KEYWORD

nice, precise, allsorted, stretch, perfect, unorderedpair, preciseworld

CONCEPT projection (info | search)

WORLD

two_segments [smaller | same | bigger]

AUTHOR

Leo Crabbe

BP555 Black and white photograph versus drawing.
(edit; present; nest [left/right]; search; history)
COMMENTS

Although the images here are related to culture (BP521left), the solution of this Bongard Problem is about the visual style of the images.

CROSSREFS

Adjacent-numbered pages:
BP550 BP551 BP552 BP553 BP554  *  BP556 BP557 BP558 BP559 BP560

KEYWORD

less, culture, experimental

WORLD

zoom in left (black_and_white_photograph)

AUTHOR

Aaron David Fairbanks

BP551 Unstable balance vs. not
(edit; present; nest [left/right]; search; history)
REFERENCE

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

CROSSREFS

Adjacent-numbered pages:
BP546 BP547 BP548 BP549 BP550  *  BP552 BP553 BP554 BP555 BP556

EXAMPLE

The classic example is a pendulum with a solid rod (instead of a string) which has a stable balance point at the bottom of its swing, where if you move the pendulum slightly it will swing back towards that balanced state. However, theoretically the pendulum can also be balanced pointing directly up. In this case, if you move the pendulum slightly it will swing down away from that upwards balanced state.

KEYWORD

updown, rotate, physics, anticomputer, perfect

CONCEPT tumbles_or_stays_put (info | search),
gravity (info | search)

AUTHOR

Jago Collins

BP548 Due to pixelation, it is ambiguous whether the middle small shape is a resized version of the left shape or the right shape vs. there is a clear choice.
(edit; present; nest [left/right]; search; history)
COMMENTS

A strange Problem relying on pixelation of digital images.

CROSSREFS

Adjacent-numbered pages:
BP543 BP544 BP545 BP546 BP547  *  BP549 BP550 BP551 BP552 BP553

KEYWORD

less, creativeexamples, pixelperfect, structure, orderedtriplet, experimental

CONCEPT imperfection_small (info | search)

WORLD

[smaller | same | bigger]

AUTHOR

Aaron David Fairbanks

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