Think you’ve cracked Martha’s puzzle from last Sunday? Reveal a solution here.
Solution to Mind the gap
The surviving labels were enough to reconstruct the whole row.
Start with the numbers we can already see.
- The 8 in position 2 forces the other 8 into position 11.
- The 6 in position 6 forces the other 6 into position 13.
- The 5 in position 9 forces the other 5 into position 15.
- The 1 in position 14 forces the other 1 into position 16.
So the row now looks like this:
| Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Channel | ? | 8 | ? | ? | ? | 6 | ? | ? |
| Position | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|
| Channel | 5 | ? | 8 | ? | 6 | 1 | 5 | 1 |
That leaves positions 1, 3, 4, 5, 7, 8, 10 and 12 for the two copies of 2, 3, 4 and 7.
Now look at the 7s. They need exactly seven other radios between them, so their positions must differ by 8.
Among the empty positions there is one possible pair:
positions 4 and 12.
So both 7s are fixed.
Now consider the 4s. They must be five positions apart. There are two possibilities:
- positions 3 and 8, or
- positions 5 and 10.
Suppose we try positions 5 and 10.
The 3s would then have to go in positions 3 and 7. But that would leave positions 1 and 8 for the two 2s, and they are much too far apart.
So that possibility cannot work.
The 4s must therefore be in positions 3 and 8.
Everything else is now forced.
The 3s go in positions 1 and 5, leaving positions 7 and 10 for the 2s.
The completed row
| Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Channel | 3 | 8 | 4 | 7 | 3 | 6 | 2 | 4 |
| Position | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|
| Channel | 5 | 2 | 8 | 7 | 6 | 1 | 5 | 1 |
giving the arrangement:
3, 8, 4, 7, 3, 6, 2, 4, 5, 2, 8, 7, 6, 1, 5, 1.
The history behind the puzzle
The original version of this problem was due to C. Dudley Langford and was published in 1958. Arrangements of this kind are now known as Langford pairings. They sometimes appear as a rule in the amazing Sudoku puzzles celebrated and discussed on one of our favourite YouTube channels, Cracking the Cryptic.
The defining rule is very simple but the problem quickly becomes too difficult to solve as the number of pairs grows. Problems like that are often too difficult for computers too, though they can usually handle much more complicated cases than humans can. This puzzle, however is a natural problem for a computer to tackle: instead of trying arrangements blindly, it can place the largest numbers first and abandon a partial arrangement as soon as it becomes impossible to complete, cutting the search space dramatically.
The combination of a tiny set of rules, some elegant human deduction and a potentially enormous search space is one reason these puzzles have remained interesting for more nearly seventy years.