In which case a barrel filled to the top with hot water will cool down faster: if you put it on ice, or if you put ice on its lid?

This question is related to the phenomenon of convection. If ice is placed in the barrel of hot water, then the lowest layer of hot water in the barrel will cool down and remain in place, further cooling of the water in the barrel will occur only due to diffusion. If you put ice on the lid of the barrel, then the upper layer of water in the barrel will cool and sink down, hot water will come in its place and the process will be repeated until the water in the barrel is completely cooled. Or the ice will melt earlier.