# Consecutive numbers that are Manhattan distance 3 apart

Can you place numbers from 1 to 16 on a 4x4 grid, such that the distance between any two consecutive numbers ($$a$$ and $$a+1$$) is Manhattan distance 3?

Bonus question: can you also make 1 and 16 be separated by Manhattan distance 3, thus making it a closed tour?

Note that the Manhattan distance between two locations is the distance between their row locations plus the distance between their column locations.

Good luck!

• If you also demand that the Euclidean distance is not 3, then this is just the classic problem of a Knight's tour on a 4x4 board. – Paul Sinclair Oct 2 at 16:38

I think this works.

Method:

It can't just be a knight's tour, since they don't exist on 4x4 boards. So I figured I would get to a corner and just knock out all four corners in a row. The rest was trial and error, except in this case I didn't happen to hit any errors.

Oh, I don't think the bonus question was in there when I was solving this. I can take a look later. Here we go:

Method:

I just backtracked from the original solution; I had to back up to the 10 and try a couple paths but it was pretty straightforward.

• Great work and very fast! Yep I made sure that it can't be a knight's tour. Let's see if you can find the bonus answer. – Dmitry Kamenetsky Oct 2 at 2:39
• Brilliant you got both of them now! – Dmitry Kamenetsky Oct 2 at 2:46

My one solutions:

10,01,08,13
07,14,11,02
04,09,16,05
15,06,03,12
with Manhattan distance 3 & closed loop in 16 & 01

• Oops, a bit slower with bonus question? :( – Conifers Oct 2 at 2:46
• Correct! Very well done. – Dmitry Kamenetsky Oct 2 at 2:47

Another solution, with bonus:

1, 14, 3, 16
4, 7, 10, 13
11, 2, 15, 6
8, 5, 12, 9

Method:

I wanted to try to get a 'symmetrical' answer - one where replacing 16 with 1, 15 with 2, etc. and flipping the 4x4 would cause the same solution. This meant I only had to 'solve' half the problem. The only way 1 and 16 AND 8 and 9 (the 'halfway' pair) could be symmetrical was with those four numbers at the four corners. It solved first guess after that.

Edit: Had to fix broken spoiler tags.

A slower solution, but my solution, with bonus.