Rebooting the Time MachineDiscrete Mathematics Level 4
The TimeWarp inc. labs have created a limited way to communicate with the past: 4 switchboards, each of which has 4 panels, each of which has 4 rows of switches, each of which has four switches. With these panels and working on a weekly-cycle, TimeWarp inc can communicate 256 bits of information back to their lab one week in the past, every week. Why not more frequently? -- because in order to make the board work, every week they need to receive an important message sent back from the future to reboot the whole apparatus (a process which takes 6 days) with one special button flipped on and the rest flipped off (the hardware was built by some "Lenova" company or something).
Which switch is needed for a successful reboot can be determined easily in the future, but using a simple code such as: "the special switch is the only one on" as the message sent back on a given week would waste ALL of the communication space for the whole week. So, every Friday, the scientists program in all 256 bits of the data that they really want to send back (sports games results, the weather on Saturday, etc.). They then want to destroy as little of that message as possible in communicating which switch needs to be off during the next reboot cycle. They've put their best scientist (you) on the task of figuring out the optimal code for sending this message back to the past each week. But how efficient can you make your code? What is the fewest number of switches you can toggle each week to communicate with your past self which switch is the reboot one? (Note: you've got the master access to toggle any switches you want, just before the final message is sent.)