With most of our staff on field operations or working at remote stations, things are occasionally quiet at BOSS front desk. One of the reception staff has developed the habit of turning whatever happens to be lying around into puzzles for the trainees, and yesterday there was particularly good material to hand. Sixteen handheld radio units had been returned after an exercise. They came in eight matched pairs, operating on channels 1 to 8, so there were two units carrying each channel number.
The units were arranged in a row according to a simple rule:
- there was exactly one other unit between the two channel 1 units;
- exactly two other units between the two channel 2 units;
- exactly three between the two channel 3 units;
- and so on, up to channel 8..
Martha was sufficiently pleased with her creation that she wrote it down to show Jodie. Unfortunately, before she got a chance to do that she spilled her coffee on the list and most of the channel labels are now unreadable. Only four survived.
The puzzle
Can you reconstruct the original row?
| Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Channel | ? | 8 | ? | ? | ? | 6 | ? | ? |
| Position | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|
| Channel | 5 | ? | ? | ? | ? | 1 | ? | ? |
Each number from 1 to 8 must appear exactly twice.
If the channel number is n, there must be exactly n other units between the two units carrying that number.
Can you find a solution
Hint 1
Start with the labels you can already see.
If one 8 is in position 2, where must the other 8 be? Do the same for the visible 6, 5 and 1.
Hint 2
Once you have placed the partners of the four surviving labels, look at the 7s.
There is only one pair of empty positions with exactly seven units between them.
Bonus challenge for programmers: can you write a short program that finds the solution and determines whether or not the four surviving labels really do determine it uniquely?
