The sixth product (part 2)

The six products are $ab, ac, ad, bc, bd$ and $cd$ . By pairing $ab$ and $cd$ , the product is $abcd$ . Likewise for pairing $ac$ and $bd$ as well as $ad$ and $bc$ . From the 5 products $2, 3, 4, 6, 9$ , there are two possibilities, namely: $2 \times 9 =3\times 6 =18$ which suggests sixth product is $\frac{18}{4}=4.5$ OR $2 \times 6 =3\times 4 =12$ which suggests sixth product is $\frac{12}{9}=\frac{4}{3}$ . So, there are $\large \textcolor{#D61F06} 2$ different values for the sixth product.

Suppose $a \le b \le c \le d$ . There are four possible values for $(a,b,c,d)$ . There are: $(a, b, c, d) =$ $\left( \frac{\sqrt{6}}{2}, \frac{2\sqrt{6}}{3}, \sqrt{6}, \frac{3\sqrt{6}}{2} \right) , \left( \frac{2\sqrt{3}}{3}, \sqrt{3}, \frac{3\sqrt{3}}{2}, 2\sqrt{3} \right) , \left( \frac{2\sqrt{2}}{3}, \sqrt{2}, \frac{3\sqrt{2}}{2},3\sqrt{2} \right) , \left( \frac{\sqrt{6}}{3}, \frac{2\sqrt{6}}{3}, \sqrt{6}, \frac{3\sqrt{6}}{2} \right)$ .