What are...Steiner systems?

Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.

This time.
What are...Steiner systems? Or: Finite geometry and puzzles.

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Slides.

Thumbnail picture.

Material used.

Steiner systems & block designs.

A bit of history.

Finite projective planes and finite geometry.

Kirkman schoolgirl problem.

Witt graph.

Applications.

Mathematica.

[弗雷克]

What if I add another constraint that every element should appear exactly m times?

[coconutart]

First of all thanks for the quick and detailed response I will utilize your kindness and knowledge for further question. Suppose I want to construct an array (6, 6, 36) S
As you explained in the video, a form of determining the number of lines and points is pp have n ^ 2 + n + 1 point and line
On each line is a number of points N + 1
This means that in the case of (6, 6, 36) S
point and line = 6^6 + 6 + 1
Number of dots placed on each line is 7?

Thank you very much for your heart junction!

[hamzaaallouche]

hey . hope u're doing great
please i have struggles to build systems S(6, 6, 36) for long term support succesfully
how can deal with it .
like the instructions on how ti build it .
thank u for u time

[luketaylor]

I've been looking to find the actual blocks of both S(5, 6, 12) and S(5, 8, 24). I've found a lot of proofs of uniqueness and stuff like that, but I want to get a hold of the actual "Block 1: (A, B, C, D, E, F) Block 2: (A, F, G...) or whatever. Anyone know where I could find that?

[coconutart]

i thanks for the great explanation!
I would love to get answered some questions to better understanding
Suppose I want to build a Steiner system for (3, 3, 9) s
For the initial 9 static of several points
For 3 mid length subgroup
For the first 3 on the left I did not understand the meaning of this entry can I get an explanation?
For the number of points (3, 3, 9) s to understand the number of points one has to perform for N ^ 2 + N + 1 I mean for this value N = 3 which means that there are 13 lines and points?
N + 1 means for this case there are 4 points in each line?

Hope I did not waste your time
Lots of congratulations
moshe

[devrimturker]

Conway felt great pressure on him to find a new thing until he discovered Mathieu M12 Leech Lattice. Then he felt more freedom on his researches.