Which cat doesn't get milk? A fun lesson in fluid dynamics

Problem: We have a tangle of tubes and four hungry cats: ginger, black, white, and grey, eagerly waiting for milk (Fig. 1). As milk begins to pour, which cat will be left without it?

Answer: Ginger, black, and white cats will not get milk.

Solution: You’ve probably seen this puzzle before, where the challenge was to find which cat gets the milk first. Some tubes were blocked, so you eliminated those, traced the open paths, and picked the winner. The puzzle's phrasing would lead you to believe there was at least a second cat that gets its share, albeit reduced. Therefore, when the puzzle's task changes to: Which cat doesn’t get milk?, you might conclude that it must be the ginger one, because its tube is blocked, while the remaining black and white cats get smaller portions. However, this is not the case. Even with an endless milk supply, the first cat will get it all, leaving none for the others.

Fig. 1. Four cats compete for the milk poured from a bottle into the network of tubes.

We can still provide for black and white cats if we slightly modify the puzzle's conditions. Some minor modifications would redirect some of the milk to the cats' tubes and partially restore justice. Breaking down this problem into two scenarios will help to grasp the basics of fluid dynamics through the puzzle's simple setup. This little riddle isn’t just cute; it’s a neat illustration of how fluid flows in real systems, such as water in plumbing or blood in arteries and veins. Let’s tackle the puzzle in its current form and then proceed to the second scenario.

Solving the puzzle as it stands. When milk is poured from the top, it follows the path of least resistance, guided by gravity (Fig. 1). The milk builds up at the bottom of Chamber 1 until it rises to the lowest outlet on the right. This outlet provides a direct path to Chamber 4, where the milk accumulates again until it reaches the lowest outlet on the right. This right tube delivers the milk directly to the grey cat's mouth.

Once milk begins to flow through that route, it never accumulates sufficiently within the network to reach the other branches. The milk finds its first way out and continues using it, ignoring all the tubs on the left and effectively cutting off other cats from the milk supply for good. In this situation, the blocked tube leading to the ginger cat doesn't affect the overall result; it might as well be unblocked.

Replacing a bottle with a tap. If you had the misfortune of leaving your bath tap running and flooding your bathroom, you know that the vent holes can't handle a strong, constant flow for too long. This happens because water keeps entering faster than it can leave. In drainage systems, it's all about a balance between the inflow and the outflow rates. The narrow vent pipe can't carry away as much water as the wider main pipe. As a result, the water in the bath rises above the vent and spills out. A steady, pressurized stream from the tap quickly outpaces what those narrow pipes can carry away.

Fig. 2. Four cats compete for the milk poured from a tap into the network of tubes.

This is precisely what we need to do to open the outlets supplying other cats. Replacing the bottle with a tap will dramatically increase the inflow, while the slimmed-down tubes will sharply decrease the outflow. When we pour gently, the puzzle behaves as it did with the bottle: milk drains through the rightmost tube, and only the grey cat drinks. But as we open the tap wider, the inflow begins to exceed what that single outlet can handle. Extra milk raises the level inside the upper chambers, gradually engaging new branches.

The first new branch to open is the one leading to the white cat. As the milk level rises inside Chamber 4, due to the stronger inflow and weaker outflow, it begins to spill into the left outlet leading to the white cat. The moment it happens, the left outlet becomes another drain, so the total outflow is now the sum of two outflows delivering milk to the grey and white cats.

If we open the tap wide, the black cat joins in. When the combined drainage of the two tubes can't handle the tap's flow, the level in Chamber 1 rises, causing milk to overflow into Chamber 2 and directly into Chamber 3. There it begins to build up, slowly rising to the ginger cat tube and spilling through if it were open. But since it's blocked, the level will continue to rise until it reaches the black cat outlet and drips into the last cat's mouth.

This staged feeding scenario is a simple demonstration of how the outflow capacity competes with the inflow rate. At a low inflow, only the grey cat receives milk. At a medium inflow, the level rises to activate the white cat’s path. At high inflow, pressure builds enough to feed the black cat. This plumbing system paints an intuitive picture of fluid dynamics at work when multiple outlets compete for the same source. Whether the next cat receives any milk depends on the strength of the pour and the diameter of the tubes, though the first cat will always receive the most. The classic puzzle becomes more intriguing when you think like a physicist and explore all possible scenarios.