# Consecutive numbers that are Manhattan distance 5 apart

Can you place numbers from 1 to 36 on a 6x6 grid, such that the distance between any two consecutive numbers ($$a$$ and $$a+1$$) is Manhattan distance 5?

Bonus question: can you also make 1 and 36 be separated by Manhattan distance 5, 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.

Here is a similar question for a 4x4 grid: Consecutive numbers that are Manhattan distance 3 apart

Good luck!

## 1 Answer

Give a non-loop solution firstly:

 04 29 16 33 36 05
13 26 23 20 11 30
32 35 08 01 14 17
19 10 03 06 27 24
22 15 12 31 34 09
07 28 25 18 21 02

Update:
Give another solution with loop:

 01 26 23 04 07 10
16 29 12 21 02 17
31 06 09 36 15 32
14 33 18 27 24 13
35 20 03 30 11 34
28 25 22 05 08 19

Strategy: We may categorize cells by distance of center:
 A B C C B A
B C D D C B
C D E E D C
C D E E D C
B C D D C B
A B C C B A

Consider that if we could make a route from A to E, and contains 1 A, 2 Bs, 3 Cs, 2 Ds and 1 E, then this is the $$1/4$$ sub-route and we could copy to another $$3/4$$ sub-routes by point symmetry.

Also E has 5 distance to A, thus we could finally connect those 4 sub-routes, forming a loop.

• I hope you don't mind the edit.. I was having a hard time seeing the numbers before.
– JS1
Commented Oct 2, 2019 at 7:17
• It's fine~ thanks! (My current environment can't upload images so I need to represent tables by all characters :( ) Commented Oct 2, 2019 at 7:35