You enter a lottery by picking a subset of three numbers from $\{1,2,3,4 \dots 14\}$ . You win a prize if you match at least two of the numbers on the winning ticket.
Show that it is possible to guarantee a win by buying $14$ tickets. (Hint: Use the $(7, 7, 3, 3, 1)$- design.)
If we make sure to have every pair appear once in the design, we are guaranteed a win.
I am not sure how this can be done using the $(7,7,3,3,1)$-design. We need $14$ elements, not $7$.
I gather we need a $(14,14,r, 3, 1)$-design, but then $r$ is not an integer.
(BTW, this is from Combinatorics by Mazur)
Either two numbers come from $1-7$, or two numbers come from $8-14$.