Math Problem of the Fortnight: Pipes
A certain industrial process requires a source of water that can be varied in temperature, but only constant-temperature water sources are available. One solution is to provide several incoming water pipes, each carrying water at some constant temperature and each having a valve that can be all the way open or all the way closed. These pipes then join into a single output pipe. The temperature of the water in the output pipe is the average (arithmetic mean) of the temperatures of the water in the incoming pipes whose valves are open.
For example, if there are three incoming water pipes carrying water at temperatures of 10°C, 20°C, and 60°C, then the following seven output temperatures can be obtained:
| Incoming pipes | ||||
|---|---|---|---|---|
| 10°C | 20°C | 60°C | ||
| Output temp. |
10°C | open | closed | closed |
| 15°C | open | open | closed | |
| 20°C | closed | open | closed | |
| 30°C | open | open | open | |
| 35°C | open | closed | open | |
| 40°C | closed | open | open | |
| 60°C | closed | closed | open | |
Your goal is to be able to obtain any output temperature from 0°C to 100°C, in one-degree increments (i.e., the 101 temperatures 0°C, 1°C, 2°C, 3°C, …, 100°C). What is the smallest number of incoming pipes needed to do this, and what temperatures of water should they carry?
(Note: It is not necessary to provide a proof that your solution is the best possible, though if you can provide such a proof, all the better.)
Winning Solution
The winning solution to this problem was by Seth Hoffert and Justin Johnson.
Another correct solution was by Rob Brase
