K. 465. A treasure trunk has an electronic lock mechanism controlled by eight switches. Every switch has two settings: on or off. The lock opens if each switch is on. It is possible to change the setting of any switch to the opposite. However, the electronic sensors will detect which switch has been manipulated, and as a result, three other switches will be automatically changed, too. (These automatic changes will not generate further switches changing.) The table below shows which switch induces which further switches to change. (For simplicity, the switches are numbered.)
|Number of switch manipulated
|Numbers of further switches changing automatically
||2, 5, 7
||1, 3, 8
||5, 6, 7
||1, 6, 8
||2, 3, 6
||2, 5, 8
||1, 3, 4
||1, 4, 7
\(\displaystyle a)\) Initially, every switch is off, except for 6 and 7. The trunk can now be opened by manually changing the setting of two appropriate switches. Which two?
\(\displaystyle b)\) Initially, every switch is off, except for 7. Is it possible to open the trunk now by manipulating the appropriate switches?
This problem is for grade 9 students only.
Deadline expired on 12 October 2015.