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# Counting question

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In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color, and at lest three of the squares are red.  How many ways are there to color the five squares?

Nov 18, 2022

#1
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There are 18 ways to color the squares.

Nov 18, 2022
#2
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In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color, and at lest three of the squares are red.  How many ways are there to color the five squares?

Since you must have three red squares, and none of them can touch each other,

then the red squares can be only squares 1, 3, and 5.

That leaves two squares to account for, i.e., squares 2 and 4.

You can color those two squares four ways, as follows:

Red   Yellow   Red   Blue   Red

Red   Blue   Red   Yellow   Red

Red   Yellow   Red   Yellow   Red

Red   Blue   Red   Blue   Red

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Nov 19, 2022
#3
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There is a contradiction in the question !

It says " so that no two consecutive squares have the same color". But, it also says " AT LEAST three of the squares are red", which means you could have 4 reds or 5 reds, which contradicts the first part that "no two consecutive squares have the same color" !

Nov 19, 2022
#4
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The statements are not contradictions. A contradiction is a combination of statements, ideas, or features of a situation that are opposed to one another. Ron’s solution is correct and satisfies the requirements of all statements, without contradiction.

Here’s an example of a contradiction:

“What you wrote is an example of the highest level of erudite intelligence.”

As you may see, what you wrote is actually an example of moronic pedantry; so my statement–presented as sarcasm, is a very obvious contradiction.

GA

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Guest Nov 19, 2022