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### 16 Squares and 5 Colors

```Date: 08/15/2002 at 18:05:23
From: Dakota Dill
Subject: Puzzle math

You have 16 squares in a box, with 5 colors: 4 blue, 3 green, 3 red,
3 white, and 3 black. The same colors can't touch or be in the same
row.

I have tried everything, using markers and every way I can, and a
color always touches.
```

```
Date: 08/15/2002 at 23:13:00
From: Doctor Peterson
Subject: Re: Puzzle math

Hi, Dakota.

Interesting puzzle! Let's try to find a way to organize our attempts,
so we can be sure we've tried everything.

Since I don't see any obvious way to solve this, I'll start by just
trying something. First, I'll need a way to show what I'm doing;
rather than use colors, I'll call the five colors X, A, B, C, D,
where there are 4 X's and 3 of each of the others:

XXXX AAA BBB CCC DDD

Since there are four X's, and there can be no more than one in any
row, there must be one in each row.

X A B C    A X B C
- - . .    - - - .
- . . .    . - . .
- . . .    . - . .

The X can go either at a corner or in the middle; and it doesn't
matter which other colors we use, as long as they are all different,
since there are the same number of each. (I haven't even said which
color is which yet.) So these are really the only two ways to start.

I marked where the X can't go, because it would be in the same column
or would touch. So there are only two places where the X could go in
the next row for the left picture, and only one for the right
picture. Let's place them, and mark where the next one can't be:

X A B C    X A B C    A X B C
- - X -    - - - X    - - - X
- - - -    - - - -    . - - -
- . - .    - - . -    . - . -

It's not hard to see where the other two X's have to go, is it?

Put those in place, then do some similar thinking about the other
colors. But for them, we have to leave each color out of one row, and
replace it with D, which has to be in three rows.

If you keep track of your thinking like this, then if you find that
you can't solve the puzzle, you can instead prove that it CAN'T be
done.

But I was able to solve it. One thing that helped was realizing that
each of the four remaining colors had to be in the middle square. See
what you can do!

Spoiler:  Here's the solution:

A X B C
C D A X
X B C D
D A X B

Note the symmetry; A, B, C, and D all have the same pattern.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
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High School Puzzles
Middle School Puzzles

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