#mathpuzzle — Public Fediverse posts

Seeing the solution here is easy, quite less easy is to be sure it's the valid solution for all 12 boxes. This could require some coding or cardboard experiments :-)

My solution to OEBP n. 850: all the boxes show the same angled 'corridor' where black-filled figures try to move through. In the left boxes the black figures are able to pass, while in the right boxes those black objects can't.

The object in box 5 (most right at the bottom) is the famous solution to the moving sofa problem:
https://en.wikipedia.org/wiki/Moving_sofa_problem
https://www.youtube.com/watch?v=rXfKWIZQIo4

#bongardproblem #mathpuzzle #puzzle #visualmath

If you think this problem is too much easy, you can write code that generates valid squares - dot pairs.

My solution to OEBP n. 818: in all 12 boxes the stroked squares seem in random positions. In the left boxes the black dot is approximately centered (in the same relative position as) where the stroked square is centered in the whole enclosing box.

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 76: both are narrowband magnitude spectrograms. In the left box the lines are wider apart. The graphs don't have scales, but the shapes of the wolvels in the two images have the same scale, so we can tell the plots have similar top frequency and time scales. So the voice in the left box is of higher frequency. We can hypothesize that there's woman voice (acute) on the left, and man voice (grave) on the right (and that's true, both are spectrograms of the spoken words "my voice").

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 75: all boxes contain magnitude spectograms of human voice. Narrow-band in the left boxes, broad-band in the right boxes.

Detailed contents:
1: Narrow-band spectrogram of the word fragment "eech" of "speech" (30 ms).
2: Narrow-band Spectrogram of a segment from "Pride and Prejudice", from:
https://courses.physics.illinois.edu/ece417/fa2019/slides/spectrograms1_wideband_narrowband.html
3: Narrow-band magnitude spectrograms of a human saying the words "my voice".
4: Narrow-band magnitude spectrograms of a human saying the words "my voice".
- - - -
7: Broad-band spectrogram of the word fragment "eech" of "speech" (5 ms).
8: Broad-band Spectrogram of a segment from "Pride and Prejudice", from:
https://courses.physics.illinois.edu/ece417/fa2019/slides/spectrograms1_wideband_narrowband.html

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 74: magnitude spectograms of human voices on the left, of other sounds or music on the right.

Detailed contents:
1: Narrow-band magnitude spectrograms of a human saying the words "my voice";
2: Ditto;
3: Narrow-band spectrogram of the word fragment "eech" of "speech" (30 ms);
4: Broad-band spectrogram of the word fragment "eech" of "speech" (5 ms);
5: Narrow-band Spectrogram of a segment from "Pride and Prejudice", from:
https://courses.physics.illinois.edu/ece417/fa2019/slides/spectrograms1_wideband_narrowband.html
6: Broad-band Spectrogram of a segment from "Pride and Prejudice";
- - - -
7: Spectrogram of great tit (Parus major) song, from Wikipedia;
8: Parts of a Chaffinch bird song;
9: Parts of a Swamp sparrow bird song;
10: The spectrogram of fragment of Erik Satie's piano piece Gymnopedie No. 1;
11: Spectrogram of a flute;
12: Spectrogram of a song from the Aphex Twin album "Windowlicker" track called "Equation".

More info:
https://hugoquene.github.io/TPhSA-EN/ch-spectrograms.html

I've used this tool for better dithering:
https://ditheringstudio.com/en/Dithering/Image

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 72: in the left boxes some regular (Archimedean) tilings in the Poincare disk model. In the right boxes, other hyperbolic tiling. An alternative simpler solution notes that in the right boxes the center of the circle is a vertex.

On the right there are tilings created by applying different Conway operators to the regular tiling {5, 4}. Images adapted from "Visual Illusions in the Hyperbolic Plane", Tuan Dung Do and Craig S. Kaplan, Bridges 2026 Conference Proceedings:
https://archive.bridgesmathart.org/2026/bridges2026-93.html

Perhaps those nice conference proceedings could be used to create some other Bongard Problems.

#bongardproblem #mathpuzzle #puzzle #visualmath

Solution of the sub-Bongard problems:
1: Like my BP61, domino-tillable or not;
2: On the left letters with an even number of strokes;
3: Like my BP62, on the left there's at least an internal node with an even number of children;
4: On the left even permutations;
5: On the left an even number of white zones;
6: On the right all graph cycles are odd;
------
7: Platonic solids vs regular filled polygons;
8: On the left at least one circle passes through the center of another;
9: On the left convex point sets;
10: Unkots vs trifoil knots (not easy);
11: Few Cube nets vs few Octahedron nets;
12: On the left two rectangles cross each other in 4 points.

So my solution BP 71: on the left there are parity-related Bongard problems.

(To keep creation time reasonable, some sub-problems are adapted from Aaron BPs and other BPs).

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 70: every box contains the representation of a planar graph (and every box on the left has a corresponding box on the right with the same graph, to make this problem simpler to solve). The graphs on the left are embedded (using various algorithms) in ways that show their symmetries and regularity more, compared to ones in the right boxes. So the embeddings on the left are generally of "higher quality".

See also:
https://spupyrev.github.io/planar-vibe/gallery.html
https://spupyrev.github.io/planar-vibe/

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 69: the left boxes show all the 6 fundamental solutions to the 7 queens problem (that has 40 solutions in total).

The n-queens puzzle is the problem of placing n chess queens on an n*n chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, column, or diagonal.

Below a visual solution too.

See also:
https://en.wikipedia.org/wiki/Eight_queens_puzzle

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 68: In the left boxes every edge belongs to at most one cycle (the graph is a cactus graph).

From Wikipedia:
"In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cacti) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle."

Below a visual solution too.

See also:
https://en.wikipedia.org/wiki/Cactus_graph

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 67: all boxes show Alpha Shapes (a generalization of Convex Hulls) of a random (some uniform, some normal) distribution of (9-30+) 2D points. I've used the nice "alphahull" package in R Studio. In the left boxes the alpha parameter is lower (2.0 - 5), while in the right boxes it's higher (5.5+) or extremely large (100, so for such datasets it becomes a proper Convex Hull).

See also:
https://en.wikipedia.org/wiki/Alpha_shape

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 66: in the left boxes the total number of runs (sub-sequences of equal color) is odd.

I think this rule is easy and quick to verify, but less intuitive to find.

This problem is simpler to understand as a 4th of this sequence by Bongard, sequences like this help with less intuitive concepts:
https://oebp.org/present.php?bp=88
https://oebp.org/present.php?bp=89
https://oebp.org/present.php?bp=90

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 65: in the left boxes the largest (stroked) object contains the center of the box.

The auxiliary lines that could help finding the solution are the diagonals of each box. If you draw them, the solution becomes much simpler to spot. Below I put the visual solution too.

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution BP 64: this BP is based on the concept of strictly Y-monotone polygons. In some boxes there are more than one figure, so in the left boxes there's a strictly Y-monotone 'ensemble' of objects.

Below I put a visual solution too.

See also:
https://en.wikipedia.org/wiki/Monotone_polygon

#bongardproblem #mathpuzzle #puzzle #visualmath

My solution: in the left boxes the support for the smallest enclosing circle is two extreme objects, in the right boxes three objects.

So those minimal circles touch two dots on the left, and three on the right. It's a computational geometry concept. Below I put a visual solution.

See also:
https://en.wikipedia.org/wiki/Smallest-circle_problem

#bongardproblem #mathpuzzle #puzzle #visualmath