The On-Line Encyclopedia of Bongard Problems

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BP230

The sum of vertices, line ends, and dots is 7 vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP225 BP226 BP227 BP228 BP229 * BP231 BP232 BP233 BP234 BP235`
	KEYWORD	`arbitrary, traditional`
	CONCEPT	`addition (info \| search),` `line_or_curve_endpoint (info \| search),` `dot (info \| search),` `vertex_of_meeting_lines (info \| search)`
	AUTHOR	`Giuseppe Insana`

BP233

Lines plus triangles equal stars vs. lines minus triangles equal stars.

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	CROSSREFS	`Adjacent-numbered pages: BP228 BP229 BP230 BP231 BP232 * BP234 BP235 BP236 BP237 BP238`
	KEYWORD	`traditional`
	CONCEPT	`addition (info \| search),` `number (info \| search)`
	AUTHOR	`Peter Shanahan`

BP288

The sum of the ratios of the filled areas is 1 vs. the sum of the ratios of the filled areas is other than 1.

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	CROSSREFS	`Adjacent-numbered pages: BP283 BP284 BP285 BP286 BP287 * BP289 BP290 BP291 BP292 BP293`
	KEYWORD	`nice, notso, traditional`
	CONCEPT	`addition (info \| search),` `outlined_filled (info \| search),` `ratio (info \| search),` `texture (info \| search)`
	AUTHOR	`"Lewis"`

BP381

Adding the top two waves yields the bottom wave vs. not so.

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	COMMENTS	`This problem is about wave interference.`
	CROSSREFS	`Adjacent-numbered pages: BP376 BP377 BP378 BP379 BP380 * BP382 BP383 BP384 BP385 BP386`
	KEYWORD	`nice, notso, math, orderedtriplet, traditional`
	CONCEPT	`addition (info \| search),` `wave (info \| search),` `2_inputs_1_output (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP790

The leftmost two add (as vectors) to the right versus no two add to a third.

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	CROSSREFS	`Adjacent-numbered pages: BP785 BP786 BP787 BP788 BP789 * BP791 BP792 BP793 BP794 BP795`
	KEYWORD	`math, structure, orderedtriplet, traditional`
	CONCEPT	`addition (info \| search),` `2_inputs_1_output (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP791

The leftmost two angles measured from thin line add to the rightmost versus no two angles add to a third.

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	COMMENTS	`This answer is independent of whether angles are measured clockwise or counter-clockwise; only a consistent choice must be made.`
	CROSSREFS	`Adjacent-numbered pages: BP786 BP787 BP788 BP789 BP790 * BP792 BP793 BP794 BP795 BP796`
	KEYWORD	`math, structure, orderedtriplet, traditional`
	CONCEPT	`addition (info \| search),` `2_inputs_1_output (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1293

Line segment separates the dots into two consecutive Fibonacci numbers vs. not so.

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	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 BP1295`
	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`

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