Some puzzles hide no trick at all, only a kind of rhythm: each number appears twice, and each number demands that its own distance be written into the row.
Langford's Tiles
Riddle statement
You have eight tiles:
1, 1, 2, 2, 3, 3, 4, 4
Arrange them in a row so that:
- there is exactly 1 tile between the two 1s;
- there are exactly 2 tiles between the two 2s;
- there are exactly 3 tiles between the two 3s;
- there are exactly 4 tiles between the two 4s.
Can it be done?
Show solution
Solution
Answer: yes.
One solution is:
$
2, 3, 4, 2, 1, 3, 1, 4
$
Check it:
- The two 1s are in positions 5 and 7. There is one tile between them: the 3.
- The two 2s are in positions 1 and 4. There are two tiles between them: 3 and 4.
- The two 3s are in positions 2 and 6. There are three tiles between them: 4, 2 and 1.
- The two 4s are in positions 3 and 8. There are four tiles between them: 2, 1, 3 and 1.
Everything fits.
Each number does more than appear twice: it forces the row to remember its value.