Question

A price draw allows you to enter by picking three numbers from {1, ..., 14}. You will win a price if you choose two of the three numbers that they will draw. Show that it is possible to guarantee a win by having 14 entries in the draw. [HINT: Use the Fano plane (7, 3, 1)-design]

Answer 1

12:44. R9 39 Title 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) +