**From New Scientist #2312, 13th October 2001** [link]

The people on the island of Fairshare have their own tax process. If *A* is the average income for the people on the island then only people earning more than *A* pay tax. If a person earns *P*, which is more than *A*, then that person pays tax (*P−A*)²/*P*.

When all the tax has been collected, then it is shared between the people earning less than *A*, in proportion to the amounts their incomes fall short of *A*.

There are 10 people on the island, *C*, *D*, *E*, *F*, *G*, *H*, *I*, *J*, *K* and *L*. Their final incomes after the tax process were:

*C* = F£117,

*D* = F£112,

*E* = F£103-58,

*F* = F£90,

*G* = F£60-47,

*H* = F£53-52,

*I* = F£51-15,

*J* = F£46-57,

*K* = F£44-20,

*L* = F£41-99 (that is to say 41 Fairshare pounds and 99 pence, where there are 116 pence to the Fairshare pound).

What were the original incomes of *E*, *H* and *K*?

See also **Enigma 1253**.

[enigma1156]

### Like this:

Like Loading...

This Python program runs in 200ms.

Solution:The original income for E was F£128. The original income for H was F£32. The original income for K was F£12.The full list of incomes before tax is as follows:

We can improve the efficiency of the program with some additional analysis. For example, the formula for the amount of tax paid means that for people earning more than A before tax the net earnings after tax are still more than A, so the only potential tax payers are C, D, E, and F.