The Transylvania Lottery picks 3 numbers from 1-14. By using two
Fano Planes, we can guarantee a match of 2 numbers.
Incidence graph
Tetrahedral Fano plane.
Notice that the Fano plane is generated by the triple $\{1, 2, 4\}$ by repeatedly adding $1$ to each entry, modulo $7$.
Steiner systems
For instance, Steiner triple systems $S(2,3,\nu)$ (the first Steiner systems studied, by Kirkman, before Steiner) exist for $\nu=0$ and all $\nu\equiv1$ or $3\allowbreak \mkern 10mu({\rm mod}\,\,6)$, and no other $\nu$.
The reverse construction, turning an $S(\tau,\kappa,\nu)$ into an $S({\tau+1},\,\allowbreak {\kappa+1},\,\allowbreak {\nu+1})$, need not be unique and may be impossible. Famously an $S(4,5,11)$ and a $S(5,6,12)$ have the Mathieu groups $M_{11}$ and $M_{12}$ as their automorphism groups, while $M_{22}$, $M_{23}$ and $M_{24}$ are those of an $S(3,6,22)$, $S(4,7,23)$ and $S(5,8,24)$, with connexions to the binary Golay code and the Leech lattice.
Fano octonions
http://math.ucr.edu/home/baez/octonions/node4.html
Perfect Code from the Fano Plan
A finite projective plane of order n is a set of n2 + n + 1 points and a set of n2 + n + 1 subsets of points called lines such that
* Any two distinct points determine a unique line
* Any two distinct lines determine a unique point
* Every point is contained in exactly n + 1 lines
* Every line contains exactly n + 1 points
A long standing unsolved problem in combinatorics is the following:
* Do there exist projective planes of order n, where n is not a prime power?
Can one hear the shape of a drum?
http://www.iop.org/EJ/article/0305-4470/38/27/L01/a5_27_l01.html
http://ej.iop.org/links/q44/eWlD3WKciX+Y6DNjNMTA1A/a5_27_l01.pdf
http://arxiv.org/PS_cache/nlin/pdf/0503/0503069.pdf
Configuration
COMMENT On this Page
The word configuration is sometimes used to describe a finite collection of points p==(p_1,...,p_n), p_i in R^d, where R^d is a Euclidean space.
The term "configuration" also is used to describe a finite incidence structure (v_r,b_k) with the following properties (Gropp 1992).
1. There are v points and b lines.
2. There are k points on each line and r lines through each point.
3. Two different lines intersect each other at most once and two different points are connected by a line at most once.
the smallest three-dimensional projective space has 15 points, 35 lines and 15 planes, with each of the planes being a Fano plane. This is a fascinating space. For instance, the 15 points can be partitioned into five disjoint lines, called a spread of the geometry, and this can be done in exactly 56 ways. It is possible to find a set of seven spreads such that each of the 35 lines is contained in exactly one of the spreads. Such a set, called a packing of the space, gives a solution to the classical Kirkman schoolgirl problem. * In fact, the 56 spreads can be partitioned into eight disjoint packings, and such a decomposition is called a hyperpacking. It turns out that our simple space-it has, after all, just 15 points-has 27,360 different hyperpackings.
http://www.findarticles.com/p/articles/mi_qa3773/is_200001/ai_n8902901
[Projective geometry] has one notable advantage: its primitive concepts are so simple that a self-contained account can be reasonably entertaining, whereas the foundations of Euclidean geometry are inevitably tedious. H. S. M. Coxeter
# A cyclic representation: the point set is {0,1,...,6} (the integers mod 7), and the triples are the set {1,2,4} of quadratic residues mod 7 and its cyclic shifts ({2,3,5}, {3,4,6}, ..., {0,1,3}).
# A binary representation: the points are all triples of binary digits except 000, and the blocks are all sets of three such triples with sum zero (for example, {110,101,011}).
# Interpreting the above as integers in base 2, we obtain the Nim representation: the point set is {1,2,...,7}, and the blocks are {1,2,3}, {1,4,5}, {1,6,7}, {2,4,6}, {2,5,7}, {3,4,7}, {3,5,6}. (These are all the winning positions in Nim consisting of three piles with at most seven counters in each.)
# Here is the famous picture of this system:
http://www.maa.org/editorial/mathgames/mathgames_07_11_05.html
http://www.gamerz.net/pbmserv/fireandice.html
http://www.otb-games.com/fireandice/rules.html