# Brainy 2

• Mar 17th 2009, 01:38 AM
u2_wa
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.
• Mar 17th 2009, 07:18 AM
Soroban
Hello, u2_wa!

This is an excellent puzzle!

Quote:

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.

• Mar 17th 2009, 07:57 AM
masters
Quote:

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  |
• Mar 17th 2009, 09:37 AM
Quote:

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 (Giggle)

Quote:

Uncountable steps after solving the problem :D in 3rd
• Mar 17th 2009, 11:22 PM
u2_wa
Quote:

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 (Giggle)

Uncountable steps after solving the problem :D in 3rd

I also did it in the second attempt, let us see what others do!!
• Mar 18th 2009, 01:27 PM
Soroban

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.