Exam 2, Part B

November 21, 1997

In this part of the exam, you may use one page of notes and a calculator.

1. (50 points) Let

N

[2]

=

N



N

,

N

[3]

=

N



(

N



N

)) and so forth. So for example,

D

n;d

=

X

[

k

]

d;d

if

k

= log

d

n

.Now, let

H

(

n; d

1

;d

2

;h;r

)=

8

>

<

>

:

X

[

h

]

d

1

;d

1



X

[

k

,

h

]

d

1

;d

1



(

X

[

h

,

r

]

d

1

;d

1



X

[

r

]

d

1

;d

2

) if

r



h

X

[

h

]

d

1

;d

1



(

X

[

k

,

r

]

d

1

;d

1



X

[

r

,

h

]

d

1

;d

2

)



X

[

h

]

d

1

;d

2

if

r



h

Note that if

d

1

=

d

2

that

H

(

n; d

1

;d

2

;h;r

)



D



n;d

1

;h

. Draw pictures of

H

16

;

2

;

3

;

2

;

1

and

H

16

;

2

;

3

;

2

;

3

.

1