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)

  • $\begingroup$ What does " [14]􏰊" mean??? $\endgroup$ – barak manos Apr 5 '16 at 5:00
  • $\begingroup$ $[n]$ is the set of integers $1,2,3, \dots, n$. $\endgroup$ – Al Jebr Apr 5 '16 at 5:00
  • $\begingroup$ Yes, but it says " [14]􏰊" :-) $\endgroup$ – joriki Apr 5 '16 at 5:01
  • $\begingroup$ $[14]=\{x \in \mathbb Z : 1\le x \le 14\}$ $\endgroup$ – Al Jebr Apr 5 '16 at 5:02
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    $\begingroup$ I guess the problem must have been that your browser didn't display the control character that barak and I were seeing. I've removed it; now it just says "[14]" as intended :-) $\endgroup$ – joriki Apr 5 '16 at 5:03

Either two numbers come from $1-7$, or two numbers come from $8-14$.

  • $\begingroup$ So, two separate sets of the $(7,7,3,3,1)$-design? $\endgroup$ – Al Jebr Apr 5 '16 at 5:31

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