Are large potatoes a better deal than small ones?

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

04 May 2008

With commodity prices rising every day, consumers have become careful about how they spend their money. This question from Lucy is a good example: “I have noticed that the size of potatoes is getting smaller. Since I buy them by the debe, am I getting cheated?”

This concern arises from the fact that when a debe is filled with potatoes, some empty spaces are left so that even though the container might be 20 litres, the actual volume occupied is less. Perhaps this is explains why market traders over-fill the debe…

Lucy’s question can therefore be rephrased this way: is the total volume of many small potatoes in a debe equal to that of few large ones? To find out, we have to study their arrangement.

If a few potatoes are poured in a debe to form a single layer at the bottom, they will form hexagonal (six-sided) patterns that tessellate to fill the circular area – most of them, anyway. The reason is that this arrangement provides the closest packing for spherical shapes.

The empty spaces in between the potatoes will also be arranged in a hexagonal pattern. Thus if a few more potatoes are poured in to form a second layer, they will also be in a similar pattern as the first group.

Continuing this way, layer by layer until the debe is full will form a very tightly packed arrangement. In the science of crystallography (the study of crystals), this pattern is called a “Hexagonal-Close-Pack” and it is quite a common configuration of atoms especially in metals. Crystallographers have calculated that perfect spheres arranged this way leave about 26 percent of the space unoccupied – regardless of the size of the individual spheres.

Unfortunately, market traders don’t have the time to pour the potatoes layer by layer and, in addition, potatoes are not perfect spheres. Consequently they never get to a hexagonal-close-pack arrangement. This means that the empty space left out is always more than the 26 percent. In crystallography, the “most wasteful” pattern leaves about 66 percent unoccupied. Thus we can estimate that the potatoes waste about 45 percent of the space in a debe.

Now, small potatoes are usually more spherical in shape than large ones. In addition, since they are lighter, the former can get to the hexagonal pattern more easily. For these reasons, the smaller potatoes waste less space in a debe – perhaps about 30 percent. Therefore, they are a better deal for the buyer.

If you want to prove this, just weigh them and compare. But that would be a boring experiment! Nonetheless, it brings out another peculiarity about Kenyan trading practices: supermarkets sell the vegetables by mass (in kilos) while kiosks measure them by volume (debes, kasukus or counting per piece). Which of the two is fairer? Well, that is a story for another day.