Infinite products with strongly $B$-multiplicative exponents

Let $N_{1,B}(n)$ denote the number of ones in the $B$-ary expansion of an integer $n$. Woods introduced the infinite product $P :=\prod_{n \geq 0} (\frac{2n+1}{2n+2})^{(-1)^{N_{1,2}(n)}}$andRobbinsprovedthat$P = 1/\sqrt{2}$. Related products were studied by several authors. We show that a trick for proving that $P^2=1\mathrm{/}2$ (knowing that $P$ converges) can be extended to evaluating new products with (generalized) strongly $B$-multiplicative exponents. A simple example is $$\prod _{n\ge 0}(\frac{Bn+1}{Bn+2}{)}^{(-1{)}^{N_{1,B}(n)}}=\frac{1}{\sqrt{B}}\mathrm{.}$$