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1.

Consider the following puzzle:

You have a rectangular chocolate bar, made up of

n

identical squares of chocolate. You can take such a bar and break it along any row or column. How many times will you have to break the bar to reduce it to

n

single chocolate squares?

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At first, this question might seem impossible. Perhaps we meant to ask for the smallest number of breaks needed. Let’s investigate.

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(a)

Suppose you started with a

1 \times 3

bar. How many breaks would you need to reduce it to single squares?

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(b)

If you had a

1 \times 4

bar, how many breaks are required?

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If you had 4 squares arranged in a

2 \times 2

square, your first break would require you to break the chocolate into two

1 \times 2

bars. Then each of these would require more break(s), for a total of breaks to go from the

2 \times 2

to single squares.

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(c)

6

-square bar could either be a

1 \times 6

bar, requiring breaks, or a

2 \times 3

bar.

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There are two ways to proceed now.

Break the bar into two $1 \times 3$ bars, each requiring more breaks, for a total of breaks.
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Break the bar into a $1 \times 2$ bar and a $2 \times 2$ bar. The $1 \times 2$ bar takes more break(s) and the $2 \times 2$ bar takes more break(s), for a total of breaks.
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