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BP384 Square number of dots vs. non-square number of dots.
(edit; present; nest [left/right]; search; history)
COMMENTS

All examples in this Problem are a collection of dots.


An equivalent solution is "Dots can be arranged into a square lattice whose convex hull is a square vs. not so". - Leo Crabbe, Aug 01 2020

CROSSREFS

Adjacent-numbered pages:
BP379 BP380 BP381 BP382 BP383  *  BP385 BP386 BP387 BP388 BP389

EXAMPLE

A single dot fits because 1 = 1*1.

A pair of dots does not fit because there is no integer x such that 2 = x*x.

KEYWORD

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

CONCEPT square_number (info | search)

WORLD

dots [smaller | same | bigger]

AUTHOR

Jago Collins

BP386 Lower shape can be used as a tile to build the upper one vs. not so.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP381 BP382 BP383 BP384 BP385  *  BP387 BP388 BP389 BP390 BP391

KEYWORD

nice, precise, allsorted, left-narrow, perfect, pixelperfect, orderedpair, traditional, preciseworld, left-listable, right-listable

CONCEPT tiling (info | search)

AUTHOR

Jago Collins

BP389 Loops are entangled (in 3-D) vs. loops can be separated (in 3-D).
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP384 BP385 BP386 BP387 BP388  *  BP390 BP391 BP392 BP393 BP394

KEYWORD

nice, precise, allsorted, contributepairs, traditional

CONCEPT knot (info | search),
topological_transformation (info | search),
imagined_motion (info | search),
motion (info | search)

WORLD

link_two_knots [smaller | same | bigger]

AUTHOR

Jago Collins

BP390 Each graph vertex is uniquely defined by its connections (the graph does not admit nontrivial automorphisms) vs. the graph admits nontrivial automorphisms.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP385 BP386 BP387 BP388 BP389  *  BP391 BP392 BP393 BP394 BP395

KEYWORD

precise, allsorted, notso, traditional, preciseworld

CONCEPT graph (info | search),
self-reference (info | search),
topological_transformation (info | search),
imagined_shape (info | search),
imagined_entity (info | search)

WORLD

connected_graph [smaller | same | bigger]

AUTHOR

Jago Collins

BP527 Each black filled circle belongs to exactly one large circle outline vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

On the left, circles define a partition of the dots.

CROSSREFS

Adjacent-numbered pages:
BP522 BP523 BP524 BP525 BP526  *  BP528 BP529 BP530 BP531 BP532

KEYWORD

precise, allsorted, traditional

CONCEPT circle (info | search),
exists_one (info | search)

AUTHOR

Aaron David Fairbanks

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

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

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

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

BP569 Triangular number of dots vs. non-triangular number of dots
(edit; present; nest [left/right]; search; history)
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

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