There are lamps, each equipped with an on/off switch. You have a remote control that allows you to change exactly switches at once. (Every time you use this remote control, you can choose which switches are to be changed.)
(a) Given that at the beginning some lamps are on, can you turn all the lamps off, using the remote control?
(b) Given that at the beginning all lamps are on, how many times do you need to use the remote control to turn all lamps off?
Let's analyze the problem step-by-step to determine the minimum number of times the remote control needs to be used to turn all lamps off, given that all lamps are initially on.
1. Initial Setup:
- There are 169 lamps, all initially on.
- The remote control can change exactly 19 switches at once.
2. First Use of the Remote:
- Turn off 19 lamps.
- Number of lamps still on: .
3. Second Use of the Remote:
- Turn off 18 lamps and turn on 1 lamp.
- Number of lamps still on: .
4. Subsequent Uses of the Remote:
- Notice that 133 is a multiple of 19: .
- Therefore, we can turn off 19 lamps at a time for the next 7 uses of the remote.
5. Calculation of Total Uses:
- First use: 1 time.
- Second use: 1 time.
- Subsequent uses: 7 times.
- Total number of uses: .
Thus, the minimum number of times the remote control needs to be used to turn all lamps off is 9.
The final answer is .