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\usepackage c a n c e l

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A4 US

1.

While wandering the halls of the math department, you find yourself staring at the captivating artwork shown below.

A square array of dots with L-shaped lines separating collections of odd numbers of dots.

(a)

How many dots are in the figure?

The dots form a by square, for a total of dots.

(b)

We can also compute the total number of dots by summing each “hook” region, from smallest to largest:

+ + + + + .

(c)

Yet another way to calculate the total number of dots is to group the terms of this sum.

1 + 11 =

;

3 + 9 =

;

5 + 7 =

.

Since there are three pairs of sums, the total is

3 \cdot

= .

(d)

If we generalize the diagram, so it has

n

hooks, how many dots will be in the largest hook?

How many dots will be in the second largest hook?

(e)

What will the sum of the smallest and largest hooks be?

What will the sum of the second smallest and second largest hooks be?

(f)

If we continue adding pairs of hooks (next smallest plus next largest), how many pairs will we have?

Multiplying then, the total number of dots will be: .

Hint.

Let’s assume that

n

is even. If it wasn’t, then there would be a single “middle” hook that isn’t added to anything, but this is counteracted by the fact that

n / 2

would count a half hook sum.