The On-Line Encyclopedia of Bongard Problems

Search: keyword:proofsrequired

Displaying 1-8 of 8 results found.

page 1

Sort: id Format: long Filter: (all | no meta | meta) Mode: (words | no words)

Tessellates the plane vs. does not tessellate the plane.

(edit; present; nest [left/right]; search; history)

	COMMENTS	`EX7152 is an example of a shape than can be stretched in such a way that it no longer tessellates the plane. This is a property that is only exhibited by shapes that tessellate with rotated copies of themselves. - Leo Crabbe, Mar 05 2021`
	CROSSREFS	`Adjacent-numbered pages: BP330 BP331 BP332 BP333 BP334 * BP336 BP337 BP338 BP339 BP340`
	KEYWORD	`nice, stretch, unstable, math, hardsort, creativeexamples, proofsrequired, perfect, pixelperfect, traditional`
	CONCEPT	`infinite_plane (info \| search),` `tessellation (info \| search),` `tiling (info \| search)`
	WORLD	`shape [smaller \| same \| bigger] zoom in left (fill_shape)`
	AUTHOR	`Aaron David Fairbanks`

BP344

Shape can tile itself vs. shape cannot tile itself.

(edit; present; nest [left/right]; search; history)

	COMMENTS	`Left examples are sometimes called "rep-tiles."` `The tiles all must be the same size. More specifically, all left examples can tile themselves only using scaled down and rotated versions of themselves with all tiles the same size. Right examples cannot tile themselves using scaled down rotated versions of themselves or even reflected versions of themselves with all tiles the same size.` `Without the puzzle piece-like shape EX4120 on the right side the current examples also allow the solution "shape can tile with itself so as to create a parallelogram vs. shape cannot tile with itself so as to create a parallelogram."`
	CROSSREFS	`See BP532 for a version with fractals.` `Adjacent-numbered pages: BP339 BP340 BP341 BP342 BP343 * BP345 BP346 BP347 BP348 BP349`
	EXAMPLE	`Go to https://oebp.org/files/yet.png for an illustration of how some left-sorted shapes tile themselves.`
	KEYWORD	`hard, nice, precise, notso, unstable, math, hardsort, creativeexamples, proofsrequired, perfect, traditional`
	CONCEPT	`recursion (info \| search),` `self-reference (info \| search),` `tiling (info \| search),` `imagined_shape (info \| search),` `imagined_entity (info \| search)`
	WORLD	`shape [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP532

Self-tiling fractal using one size of tile vs. does not tile itself with a single size of itself.

(edit; present; nest [left/right]; search; history)

	CROSSREFS	`This is BP344 ("rep-tiles") but for fractals.` `See BP1119 for the version with multiple different sizes of tile allowed.` `Adjacent-numbered pages: BP527 BP528 BP529 BP530 BP531 * BP533 BP534 BP535 BP536 BP537`
	KEYWORD	`hardsort, proofsrequired, perfect, infinitedetail, contributepairs`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search),` `tiling (info \| search)`
	WORLD	`[smaller \| same \| bigger] zoom in left (fractal_self_tile)`
	AUTHOR	`Aaron David Fairbanks`

BP850

Shape can be maneuvered around the corner vs. not so.

(edit; present; nest [left/right]; search; history)

	REFERENCE	`https://en.wikipedia.org/wiki/Moving_sofa_problem`
	CROSSREFS	`Adjacent-numbered pages: BP845 BP846 BP847 BP848 BP849 * BP851 BP852 BP853 BP854 BP855`
	KEYWORD	`nice, precise, physics, creativeexamples, proofsrequired, left-narrow, right-narrow, dithering`
	CONCEPT	`rotation_required (info \| search),` `imagined_motion (info \| search),` `physically_fitting (info \| search)`
	AUTHOR	`Leo Crabbe`

BP1119

Tiled by finitely many smaller copies of itself (different sizes allowed) vs. not so.

(edit; present; nest [left/right]; search; history)

	COMMENTS	`These are sometimes called "irreptiles".`
	CROSSREFS	`See BP532 for the version with only one size of tile allowed.` `Adjacent-numbered pages: BP1114 BP1115 BP1116 BP1117 BP1118 * BP1120 BP1121 BP1122 BP1123 BP1124`
	KEYWORD	`hardsort, proofsrequired, perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search),` `tiling (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP1137

Constructible Polygon vs. Non-constructible Polygon

(edit; present; nest [left/right]; search; history)

	REFERENCE	`https://en.wikipedia.org/wiki/Straightedge_and_compass_construction` `https://en.wikipedia.org/wiki/Constructible_polygon`
	CROSSREFS	`Adjacent-numbered pages: BP1132 BP1133 BP1134 BP1135 BP1136 * BP1138 BP1139 BP1140 BP1141 BP1142`
	KEYWORD	`stub, precise, math, hardsort, proofsrequired, preciseworld`
	AUTHOR	`Jago Collins`

BP1200

The whole rectangle can be filled in by successively replacing pairs of adjacent rectangles with one vs. not so.

(edit; present; nest [left/right]; search; history)

	COMMENTS	`Another wording: "can be repeatedly broken along 'fault lines' to yield individual pieces vs not."`
	REFERENCE	`Robert Dawson, A forbidden suborder characterization of binarily composable diagrams in double categories, Theory and Applications of Categories, Vol. 1, No. 7, p. 146-145, 1995.`
	CROSSREFS	`All of the examples fitting left here would fit right in BP1199 except for (1) a single rectangle, (2) two rectangles stacked vertically, or (3) two rectangles side by side horizontally.` `All of the examples fitting right in in BP1097 (re-styled) would fit right here (besides a single solid block, but that isn't shown there).` `Adjacent-numbered pages: BP1195 BP1196 BP1197 BP1198 BP1199 * BP1201 BP1202 BP1203 BP1204 BP1205`
	KEYWORD	`hard, precise, challenge, proofsrequired, inductivedefinition, left-listable, right-listable`
	AUTHOR	`Aaron David Fairbanks`

BP1245

When two players alternate coloring regions, either can force connection from top edge to bottom edge vs. either can force connection from left edge to right edge.

(edit; present; nest [left/right]; search; history)

	CROSSREFS	`Adjacent-numbered pages: BP1240 BP1241 BP1242 BP1243 BP1244 * BP1246 BP1247 BP1248 BP1249 BP1250`
	KEYWORD	`hard, precise, convoluted, dual, rotate, boundingbox, hardsort, challenge, proofsrequired`
	AUTHOR	`Aaron David Fairbanks`

page 1

		Hints
(Greetings from The On-Line Encyclopedia of Bongard Problems!)