We study lower bounds for dyadic square functions of indicator functions. In the case of the dyadic square function $S_{2}$ we obtain a sharp lower bound: for every measurable $A \subset {[0,1)}$, we have \[ \|S_{2}(\mathbbm{1}_{A})\|_{1}\ge \mathbb{E}_{|A|}\big[\sqrtτ\big]\asymp |A|^{*}\log_2\frac{1}{|A|^{*}}, \] where $τ$ is the first exit time from $(0,1)$ of a standard Brownian motion started at $|A|$, and $|A|^{*}:=\min\{|A|,1-|A|\}$. This estimate gives logarithmic improvement over the classical Burkholder--Davis--Gundy lower bound $|A|^{*}$. In addition, we show a sharp inequality \[ \|S_{1}(\mathbbm{1}_{A})\|_{1} \ge T(|A|)\asymp |A|^{*}\log_{2}\frac{1}{|A|^{*}}, \] where $T(x)=\sum_{k=0}^{\infty}\frac{\operatorname{dist}(2^{k}x,\mathbb{Z})}{2^{k}}$ is the Takagi function.