We extend the Keynesian Beauty Contest game to a second dimension. In this new, «planar beauty contest» game, groups of subjects are asked to simultaneously form guesses about two variables, X and Y, each in a closed interval [0, 100]. Average guesses matter for the realizations of the two target values for X and Y that subjects are incentivized to achieve, i.e., the system of equations is coupled. Such coupled systems represent a workhorse, reduced-form framework for many models in macroeconomics and other fields.