adamnamsole
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Side by side square
- Thread starter adamnamsole
- Start date
Dr.Peterson
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Clearly this is meant as a chance to stretch your own thinking abilities, so any help I give would only prevent you from reaching that goal.
My best advice is to TRY SOMETHING. You'll probably discover something to look out for that can go wrong. (E.g., the two middle squares are hardest to fill in, so you might want to start there. I learned that by starting at an end.) Once you've learned something about the way the problem works, TRY SOMETHING ELSE. This is all about learning as you go.
If you had more experience, you might come at this with some ideas already. But those ideas get into your mind as a result of having gone through this work before for similar puzzles. So milk this problem for all it's worth!
adamnamsole
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The thing is I don’t understand what they’re trying to say the question is arrange the numbers 1 to 8 in the squares so that No two consecutive integers touch at a side or a vertex (corner)
Following Dr. Peterson's advice, lets START by labelling the cells a and b going from left to right in the top row, c, d, e, and f going from left to right in the middle row, and g and h going from left to right in the bottom row.
Now consider placing one of the integers other than 1 or 8 in cell d. Call it integer n. Where do you place integer n + 1? If you place n + 1 in cells a, c, e, or g, that cell containing n + 1 touches cell d containing n along an edge. But n and n + 1 are successive integers, and their cells must not touch. If you place n + 1 in cell b or h, that touches cell d at a corner. Therefore, you must place n + 1 in cell f. But then where can you place n - 1? No cell works. The same logic applies to cell e. Therefore, 1 and 8 must go in cells d and e. Let's try 1 in cell d and 8 in cell e. But then 2 must go in cell f and 7 must go in cell c. Now what?
Now consider placing one of the integers other than 1 or 8 in cell d. Call it integer n. Where do you place integer n + 1? If you place n + 1 in cells a, c, e, or g, that cell containing n + 1 touches cell d containing n along an edge. But n and n + 1 are successive integers, and their cells must not touch. If you place n + 1 in cell b or h, that touches cell d at a corner. Therefore, you must place n + 1 in cell f. But then where can you place n - 1? No cell works. The same logic applies to cell e. Therefore, 1 and 8 must go in cells d and e. Let's try 1 in cell d and 8 in cell e. But then 2 must go in cell f and 7 must go in cell c. Now what?
Dr.Peterson
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Okay, let me show you my first try, and talk through it.
Here's the grid:
. _ _ .
_ _ _ _
. _ _ .
Suppose I start with 1 on the left; here are the places I can put the 2:
. _ 2 .
1 _ 2 2
. _ 2 .
The three blanks are where I can't put it, because 1 and 2 are consecutive, and those squares touch my 1 along an edge or at a corner. Okay so far?
I'll pick a place to put my 2, and mark where the 3 can go:
. _ 2 .
1 _ _ _
. 3 3 .
All those blanks touch my 2 along an edge or at a corner. Right?
I'll pick a place for the 3, and show where the 4 can go:
. 4 2 .
1 _ _ 4
. 3 _ .
See if you can continue from there. You'll see what goes wrong, and learn something from it, namely that those middle squares need to be used up fairly early.
pka
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Sorry about the LaTeX, but the gives a basic idea. Please post others.
\(\begin{array}{*{20}{c}} {}&{\boxed{\;4\;}}&{\boxed{\;6\;}}&{} \\ {\boxed{\;7\;}}&{\boxed{\;1\;}}&{\boxed{\;8\;}}&{\boxed{\;2\;}} \\ {}&{\boxed{\;3\;}}&{\boxed{\;5\;}}&{}\end{array}\)
\(\begin{array}{*{20}{c}} {}&{\boxed{\;4\;}}&{\boxed{\;6\;}}&{} \\ {\boxed{\;7\;}}&{\boxed{\;1\;}}&{\boxed{\;8\;}}&{\boxed{\;2\;}} \\ {}&{\boxed{\;3\;}}&{\boxed{\;5\;}}&{}\end{array}\)
Otis
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Hi adam. You can replace the word "integers" with "numbers". The numbers are:
1 2 3 4 5 6 7 8
If two numbers are next to each other in that list, then they are not allowed to be placed in boxes that are touching each other (even if the boxes touch only at a corner). We solve puzzles like this using trial and error, eventually recognizing patterns or coming to realizations that lead us to a solution. That, or we just get lucky.
