Solution to my BP n.18: boxes on the left represent Catalan number series in various ways. The boxes on the right represent other well known series.

A nice book on the topic is "Catalan Numbers" by Richard P. Stanley (2015).

Detailed contents of each box:
1: distinct ways to triangulate (connecting vertices with non-crossing line segments) a regular polygon with n-2 sides (here n=4) in n triangles (C4);
2: all different "mountain ranges" with n upstrokes and n downstrokes, starting and ending at zero height, never going below zero. Here n=3 (C3);
3: about the same, rotated (C3);
4: all 14 differently shaped binary trees with four nodes (still C4).
5: Catalan numbers sequence up to 42;
6: all 14 different complete parenthesizations of 5 black balls (C4);
- - - -
7: all 22 Young diagrams with 8 (unlabeled) boxes (https://en.wikipedia.org/wiki/Young_tableau , also unlabeled partition numbers: https://oeis.org/A000041 );
8: all 16 binary numbers with 4 digits (powers of two);
9: Fibonacci numbers sequence up to 34;
10: all 16 binary numbers with 4 digits in a Gray order (https://en.wikipedia.org/wiki/Gray_code );
11: all permutations of 4 different items (factorials);
12: 4 stars (*), 3 urns, 2 separators (|). C(6,4) = C(6,2) = 15 ways (triangular numbers, n choose 2).

Boxes 1 and 11 created with little Python scripts and the Pillow library. The others are pixel art by hand. Box 6 created with both.

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

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Unlike most others, in this Bongard problem the boxes on the left are approximately paired with the right ones to make it simpler to find (to see) the solution. The boxes appear to contain some random 1D distributions (on the other axis they are always uniformly scattered). The distributions have slightly different densities from left to right, but they seem the same or very similar. And the cursors seem to point to some measure of central tendency. But the cursors are in slightly different positions between the two sides, consistently. Once you remember one of the first images in statistics books, you're done.

https://en.wikipedia.org/wiki/File:Visualisation_mode_median_mean.svg

Solution to my BP n.17: In the left boxes the cursor at the base points approximately at the medians of the distributions. While in the right boxes they point to their mean.

Detailed contents of each box:
1: Uniform distribution;
2: Gaussian distribution centred in the middle of the box and with sigma=20;
3: Triangular distribution, the mean is shifted 26 pixels on the right of middle of the box;
4: Exponential distribution with lambda=0.03;
5: Gamma distribution with alpha=6.0 and beta=6.0;
6: (empty);
- - - -
7: distribution like box 1;
8: distribution like box 2;
9: distribution like box 3;
10: distribution like box 4;
11: distribution like box 5;
12: (empty).

All of them are famous distributions. In each box I've plotted about 1000-2000 dots, but the amount varies between boxes. I've generated the boxes using small Python programs using the real-valued-distributions of the Python standard library and the Pillow library. Means and medians found analytically where possible, numerically otherwise.

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Solution to my BP n.16: in the left boxes links (made of two threads), vs knots (made of one thread) in the right boxes.

Images adapted (and fixed) from:
https://aeb.win.tue.nl/at/algtop-5.html

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Solution to my BP n.15: boxes on the left contain connected Julia sets, where "c" is part of the Mandelbrot set (while the boxes on the right contain unconnected Julia sets, where "c" is outside the Mandelbrot set. Fatou dust).

In box 4 there are some unconnected pixels, so it's an approximation. Some Bongard problems are designed to be pixel-perfect, some aren't.

c coordinates for each box (of the improved problem version):
1: c = -0.5170035144820032 + 0.5039938589126339i;
2: c = -0.2553481329691844 - 0.6474769737617956i;
3: c = -0.1255630077892574 + 0.6562411957352091i;
4: c = -0.16942824654638486 + 0.80775223256367i;
5: c = -0.6762463320476096 - 0.14260685206116855i;
6: c = 0.3520525517735108 + 0.240080705507128i;
- - - -
7: c = -0.684477263178615 + 0.5231576421224597i;
8: c = -0.18974882300362994 - 1.0000662124099033i;
9: c = -1.1893363393831355 - 0.37942348445846i;
10: c = 0.3279450230800282 + 0.046672932419438665i;
11: c = -0.3057398133377645 - 0.8042142733599712i;
12: c = -1.8693622366487037 - 0.05255728368661179i

For more info:
https://en.wikipedia.org/wiki/Julia_set

This time I was lazier so I have just adapted the nice Julia sets images from @fractalbot by Philipp Joram on Mastodon.

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