# Transylvania lottery

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The Fano plane

In mathematical combinatorics, the Transylvanian lottery is a lottery where three numbers between 1 and 14 are picked by the player for any given ticket, and three numbers are chosen randomly. The player wins if two of their numbers, on a given ticket, are among the random ones. The problem of how many tickets the player must buy in order to be certain of winning can be solved by the use of the Fano plane. (Mazur 2010, p.280 problem 15) (Javier Martínez, Gloria Gutiérrez & Pablo Cordero et al. 2008, p.85)

The solution is to buy a total of 14 tickets, in two sets of seven. One set of seven is every line of a Fano plane with the numbers 1-7, the other with 8-14, i.e.:

1-2-5, 1-3-6, 1-4-7, 2-3-7, 2-4-6, 3-4-5, 5-6-7, 8-9-12, 8-10-13, 8-11-14, 9-10-14, 9-11-13, 10-11-12, 12-13-14.

Because at least two of the winning numbers must be either high (8-14) or low (1-7), and every high and low pair is represented by exactly one ticket, you would be guaranteed at least two correct numbers on one ticket with these 14 purchases. 21/26 of the time you will have one ticket with two numbers matched. If all three winning numbers are either high or low you would either have one ticket with all three numbers (1/26 chance of this occurring), or three different tickets that each matched two (4/26 chance).

## References

• Martínez, Javier; Gutiérrez, Gloria; Cordero, Pablo; Rodríguez, Francisco J.; Merino, Salvador (2008), "Algebraic topics on discrete mathematics", in Moore, Kenneth B. (ed.), Discrete mathematics research progress, Hauppauge, NY: Nova Sci. Publ., pp. 41–90, ISBN 978-1-60456-123-4, MR 2446219
• Mazur, David R. (2010), Combinatorics, MAA Textbooks, Mathematical Association of America, ISBN 978-0-88385-762-5, MR 2572113