The dishes Problem

 My first step was to reword the problem and understand what was happening. When the chef says “every 2 used a dish of rice” it means that each guest had half a dish of rice. Another way of looking at this is that, of the total number of guests, half of them had a dish of rice, one third had a dish of broth, and one quarter had a dish of meat.

Let’s represent the total number of guests by twelve groups of one twelfth. This is because we can easily figure out halves, thirds, and quarters of twelve. Half the guests had a rice dish, that is 6 twelfths. A third had a broth dish, that is four twelfths. A quarter had a meat dish, that is three twelfths.

Rice

Rice

Rice

Rice

Rice

Rice

Broth

Broth

Broth

Broth

Meat

Meat

Meat

 

You can see that we have an extra twelfth of dishes. Because there are 65 dishes served, there is an extra twelfth. The problem becomes “What number, when increasing that number by one twelfth of itself, becomes 65.” Traditionally, algebra would be used at this point. However, a culture could guess and check until they arrived at the correct number of 60 guests.


While presenting itself as a problem of Chinese origin, the Chinese culture does not exist in the problem outside of the choices of food. However, the existence of the problem as a piece of history, going so far as to provide the original historical text, is very important. It shows that the traditional western view of mathematics is incomplete by giving a concrete example of maths existing outside that realm.

The fact that this is presented as a story does make the problem more enjoyable to solve, almost as if we are being transported back to an ancient Chinese restaurant. It is still nonsensical, one would simply count the number of tables and average table size before counting dishes. But if one can understand the problem as a fun test of our math abilities with a flair of the dramatic, it becomes much more enjoyable.