We present here a method for computational color constancy in which a deep convolutional neural network is trained to detect achromatic pixels in color images after they have been converted to grayscale. The method does not require any information about the illuminant in the scene and relies on the weak assumption, fulfilled by almost all images available on the web, that training images have been approximately balanced. Because of this requirement we define our method as quasi-unsupervised. After training, unbalanced images can be processed thanks to the preliminary conversion to grayscale of the input to the neural network. The results of an extensive experimentation demonstrate that the proposed method is able to outperform the other unsupervised methods in the state of the art being, at the same time, flexible enough to be supervisedly fine-tuned to reach performance comparable with those of the best supervised methods.