5
$\begingroup$

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!

$\endgroup$
6
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ I hope you don't mind the edit.. I was having a hard time seeing the numbers before. $\endgroup$ – JS1 Oct 2 at 7:17
  • $\begingroup$ It's fine~ thanks! (My current environment can't upload images so I need to represent tables by all characters :( ) $\endgroup$ – Conifers Oct 2 at 7:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.