In mathematical combinatorics, the Transylvania lottery is a lottery where players selected three numbers from 1-14 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. (Javier Martínez, Gloria Gutiérrez & Pablo Cordero et al. 2008, p.85)(Mazur 2010, p.280 problem 15)
The Fano plane with points labelled
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.
Low set
1-2-5
1-3-6
1-4-7
2-3-7
2-4-6
3-4-5
5-6-7
High set
8-9-12
8-10-13
8-11-14
9-10-14
9-11-13
10-11-12
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 (6/13)*(5/12)=5/26 chance that all three randomly chosen numbers are in the same Fano plane set. In this case, there is a 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.
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, ISBN978-1-60456-123-4, MR 2446219
transylvania, lottery, mathematical, combinatorics, lottery, where, players, selected, three, numbers, from, each, ticket, then, three, numbers, chosen, randomly, ticket, wins, numbers, match, random, ones, problem, asks, many, tickets, player, must, order, ce. In mathematical combinatorics the Transylvania lottery is a lottery where players selected three numbers from 1 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 Javier Martinez Gloria Gutierrez amp Pablo Cordero et al 2008 p 85 Mazur 2010 p 280 problem 15 The Fano plane with points labelledAn 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 Low set 1 2 5 1 3 6 1 4 7 2 3 7 2 4 6 3 4 5 5 6 7High set 8 9 12 8 10 13 8 11 14 9 10 14 9 11 13 10 11 12 12 13 14At 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 6 13 5 12 5 26 chance that all three randomly chosen numbers are in the same Fano plane set In this case there is a 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 editCombinatorial design Lottery WheelingReferences editMartinez Javier Gutierrez Gloria Cordero Pablo Rodriguez 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 nbsp This combinatorics related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Transylvania lottery amp oldid 1170516613, wikipedia, wiki, book, books, library,
