Making connections
Clearing out a cupboard at BOSS Technical Stores has unearthed a box of old signal adaptors from one of our legacy field systems.
The system used six different kinds of connection, labelled A to F. Each adaptor joins one type of connection to another and can be used either way round.
There are twelve adaptors in the box:
| A – D | B – E | C – F | A – E |
| B – D | D – F | B – C | E – F |
| A – F | C – E | C – D | B – F |
The old service manual says that the diagnostic equipment could test the entire set in a single run by joining all twelve adaptors into one continuous chain.
There is only one rule: adjoining ends must carry the same letter.
So, for example, an A – D adaptor could be followed by D – F, which could then be followed by F – C.
Can you arrange all twelve adaptors into a single unbroken chain? You must use every adaptor exactly once. You may turn any adaptor round.There is also a second question.
What letters must appear at the two free ends of your finished chain? Can you work that out before you find the chain?
Hint 1
Count how many times each of the letters A, B, C, D, E and F appears among the twelve adaptors.
Hint 2
Think about any letter that appears somewhere inside the completed chain.
Every time the chain arrives at a connection of that type, it must also leave through another one. What does that tell you about letters that occur an odd number of times?
Bonus challenge: could the twelve adaptors ever be joined into a closed loop, with no free ends at all? Explain your answer.