Let $S = A_1 \cup A_2 \cup \cdots \cup A_6 = B_1 \cup B_2 \cup \cdots \cup B_n$. Find $n$ if each element of $S$ belongs to four $A$s and $3$ $B$s.

Question

The six sets $A_1, \ldots, A_6$ each contain $4$ elements. The $n$ sets $B_1, \ldots, B_n$ each contain $2$ elements. Let $S = A_1 \cup A_2 \cup \cdots \cup A_6 = B_1 \cup B_2 \cup \cdots \cup B_n$. Each element of $S$ belongs to exactly four of the sets $A_1, \ldots, A_6$ and exactly three of the sets $B_1, \ldots, B_n$. Find $n$.

Answer 1

Say $S =\{1,2,3,...,k\}$. Then $S = A_1 \cup A_2 \cup \cdots \cup A_6$ we have $$6\cdot 4 = k\cdot 4 \implies k=6$$

From $S = B_1 \cup B_2 \cup \cdots \cup B_n$ we have $$n\cdot 2 = k\cdot 3\implies n = 9$$

Answer 2

Strategy:

1. Given that the $6$ sets $A_1, \ldots, A_6$ each contain $4$ elements, how many elements would there be in their union if they were disjoint?
2. Given that each element of $S$ belongs to exactly $4$ of the $6$ sets $A_1, \ldots, A_6$, how many elements are in $S$?
3. Given that the $n$ sets $B_1, \ldots, B_n$ each contain $n$ elements, how many elements would there be in their union if they were disjoint?
4. Given that each element of $S$ belongs to exactly $3$ of the $n$ sets $B_1 \ldots, B_n$, how many elements are in the union?
5. Equate the results of steps 2 and 4 to solve for $n$.

Answer 3

Let $$S=\{1,2,3,4,5,6\}\\A_1=\{1,2,3,4\}\\A_2=\{2,3,4,5\}\\A_3=\{3,4,5,6\}\\A_4=\{4,5,6,1\}\\A_5=\{5,6,1,2\}\\A_6=\{6,1,2,3\}$$therefore $$B_1=\{1,2\}\\B_2=\{1,4\}\\B_3=\{1,6\}\\B_4=\{2,4\}\\B_5=\{2,5\}\\B_6=\{3,4\}\\B_7=\{3,5\}\\B_8=\{3,6\}\\B_9=\{5,6\}\\$$