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[[File:Fano_plane_lottery.svg|thumb|The Fano plane with points labelled]]
In mathematical combinatorics, the '''Transylvania lottery''' is a [[lottery]] where players selected three numbers from 1 to 14 for each ticket, and then three numbers are chosen randomly. A ticket wins if two of the numbers match the random ones. The problem asks how many tickets the player must buy in order to be certain of winning.
{{harvs | last1=Martínez | first1=Javier | last2=Gutiérrez | first2=Gloria | last3=Cordero | first3=Pablo | last4=Rodríguez | first4=Francisco J. | last5=Merino | first5=Salvador | editor1-last=Moore | editor1-first=Kenneth B. | title=Discrete mathematics research progress | url=https://books.google.com/books?id=_uCQVCAXOJAC&pg=PA85 | publisher=Nova Sci. Publ. | location=Hauppauge, NY | isbn=978-1-60456-123-4 |mr=2446219 | year=2008 | chapter=Algebraic topics on discrete mathematics | pages=41–90|loc=p.85}}{{harv|Mazur|2010|loc=p.280 problem 15}}

An upper bound can be given using the [[Fano plane]] with a collection of 14 tickets in two sets of seven. Each set of seven uses every line of a Fano plane, labelled with the numbers 1 to 7, and 8 to 14.

{| class="wikitable" style="font-size:125%; font-weight:bold; text-align:center; background:white;"
! style="font-size:80%;"|Low set
| {{red|1-2-5}} || {{blue|1-3-6}} || {{fuchsia|1-4-7}} || {{black|2-3-7}} || {{cyan|2-4-6}} || {{orange|3-4-5}} || {{green|5-6-7}}
|-
! style="font-size:80%;"|High set
| {{red|8-9-12}} || {{blue|8-10-13}} || {{fuchsia|8-11-14}} || {{black|9-10-14}} || {{cyan|9-11-13}} || {{orange|10-11-12}} || {{green|12-13-14}}
|}

At least two of the three randomly chosen numbers must be in one Fano plane set, and any two points on a Fano plane are on a line, so there will be a ticket in the collection containing those two numbers. There is a {{sfrac|6|13}}×{{sfrac|5|12}}={{sfrac|5|26}} chance that all three randomly chosen numbers are in the same Fano plane set. In this case, there is a {{sfrac|1|5}} chance that they are on a line, and hence all three numbers are on one ticket, otherwise each of the three pairs are on three different tickets.

==See also==
* [[Combinatorial design]]
* [[Lottery Wheeling]]

==References==

*{{Citation | last1=Martínez | first1=Javier | last2=Gutiérrez | first2=Gloria | last3=Cordero | first3=Pablo | last4=Rodríguez | first4=Francisco J. | last5=Merino | first5=Salvador | editor1-last=Moore | editor1-first=Kenneth B. | title=Discrete mathematics research progress | url=https://books.google.com/books?id=_uCQVCAXOJAC&pg=PA85 | publisher=Nova Sci. Publ. | location=Hauppauge, NY | isbn=978-1-60456-123-4 |mr=2446219 | year=2008 | chapter=Algebraic topics on discrete mathematics | pages=41–90}}
*{{Citation | last1=Mazur | first1=David R. | title=Combinatorics | url=https://books.google.com/books?id=yI4Jx5Obr08C&pg=PA280 | publisher=[[Mathematical Association of America]] | series=MAA Textbooks | isbn=978-0-88385-762-5 |mr=2572113 | year=2010}}


[[Category:Combinatorics]]


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