1. ## Brainy 2

Fit digits 1 to 8 in such a way that no two consecutive numbers are next to each other.
An example is also attached.

2. Hello, u2_wa!

This is an excellent puzzle!

Place the digits 1 to 8 so that no two consecutive numbers are adjacent.
("Adjacent" numbers are in cells that share a side or a vertex.)
Code:
          *---*
|   |
*---*---*---*
|   |   |   |
*---*---*---*
|   |   |   |
*---*---*---*
|   |
*---*

I solved this while in college (back in the Jurassic Period)
. . and have a logical solution.

I'll post it later, after everyone has tried it.

3. Originally Posted by u2_wa
Fit digits 1 to 8 in such a way that no two consecutive numbers are next to each other.
An example is also attached.
Hi u2_wa,

I have no logical approach to this one, but through trial and many errors, I came up with this. I'll 'white' it out, in case others are still struggling and want to solve it.

Code:

| 2  |
--------------
| 6 | 8  | 5 |
--------------
| 4 | 1  | 3 |
--------------
| 7  |

4. Originally Posted by masters
Hi u2_wa,

I have no logical approach to this one, but through trial and many errors, I came up with this. I'll 'white' it out, in case others are still struggling and want to solve it.

Code:

| 2  |
--------------
| 6 | 8  | 5 |
--------------
| 4 | 1  | 3 |
--------------
| 7  |
Believe it or not I got that solution on 2nd or 3rd try but unfortunately I was searching for the number of solutions to this problem before I read your posts

Uncountable steps after solving the problem in 3rd

Believe it or not I got that solution on 2nd or 3rd try but unfortunately I was searching for the number of solutions to this problem before I read your posts

Uncountable steps after solving the problem in 3rd
I also did it in the second attempt, let us see what others do!!

6. Okay, here's my "logical" approach to this puzzle.

We have the eight cells labeled $\displaystyle a$ to $\displaystyle h$.
There are two "central cells", $\displaystyle c$ and $\displaystyle h.$
Code:
          *---*
| a |
*---*---*---*
| b | c | d |
*---*---*---*
| e | f | g |
*---*---*---*
| h |
*---*

There are eight digits to install:
. . two "end" numbers and six "middle" numbers.

. . $\displaystyle \underbrace{1}_{\text{end}},\underbrace{2,3,4,5,6, 7}_{\text{midde}},\underbrace{8}_{\text{end}}$

Place a middle number in a central cell, say, 4 in cell $\displaystyle c$.
Code:
          *---*
| x |
*---*---*---*
| x | 4 | x |
*---*---*---*
| x | x | x |
*---*---*---*
| h |
*---*

We see that its neighbors (3 and 5) cannot be placed
. . in cells $\displaystyle a,b,d,e,f,g.$
So both 3 and 5 cannot be installed.

Conclusion: the central cells must not contain a middle number.

Hence, 1 and 8 must go in cells $\displaystyle c$ and $\displaystyle f.$

Code:
          *---*
| a |
*---*---*---*
| b | 1 | d |
*---*---*---*
| e | 8 | g |
*---*---*---*
| h |
*---*

2 cannot be in cells $\displaystyle a,b,d,e,g \quad\Rightarrow\quad 2 \in h.$

7 cannot be in cells $\displaystyle b,d,e,g \quad\Rightarrow\quad 7 \in a.$
Code:
          *---*
| 7 |
*---*---*---*
| b | 1 | d |
*---*---*---*
| e | 8 | g |
*---*---*---*
| 2 |
*---*

3 cannot be in $\displaystyle e$ or $\displaystyle g$ . . . 3 may be in $\displaystyle b$ or $\displaystyle d.$

Pick one: say, $\displaystyle 3 \in b$
Code:
          *---*
| 7 |
*---*---*---*
| 3 | 1 | d |
*---*---*---*
| e | 8 | g |
*---*---*---*
| 2 |
*---*

6 cannot be in $\displaystyle d.$

Also, 6 cannot be in cell $\displaystyle e.$
Else 4 and 5 will be in cells $\displaystyle d$ and $\displaystyle g$ (and be adjacent).
. . Hence: .$\displaystyle 6 \in g.$
Code:
          *---*
| 7 |
*---*---*---*
| 3 | 1 | d |
*---*---*---*
| e | 8 | 6 |
*---*---*---*
| 2 |
*---*

4 cannot be in cell $\displaystyle e$ . . . $\displaystyle 4 \in d$

And finally: .$\displaystyle 5 \in e$
Code:
          *---*
| 7 |
*---*---*---*
| 3 | 1 | 4 |
*---*---*---*
| 5 | 8 | 6 |
*---*---*---*
| 2 |
*---*

Solution

Disregarding rotations and reflections, there is one solution.