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# Now, it is simple to verify that

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(1)

(2)

22

2

2

ott

2

3

2

2

2

2

2

2

2

2

2

2

2

2

2

2

(3)

ott

2

2

2

3

2

2

2

2

2

2

2

2

2

2

2

2

2

2

(4)

−i2ϕ

j

2

j2

j

4

−i2ϕ

j

2

j2

−i2ϕ

j

2

2j

−i2ϕ

j

2

j2

π2

j

2

2j

j

2

j2

j

2

2j

j

2

j2

(5)

T

T

T

T

T

T

T

T

T

T

T

(6)

2

πj(h−k)n

πj(h+k)n

2

πj(h−k)n

πj(h+k)n

2

2

iπj(h−k)

iπj(h−k)n

iπj(h+k)

iπj(h+k)n

2

iπj(h−k)n

iπj(h+k)n

2

2

(7)

1

n

1

2

1

3

1

4

2

22

1

2

24

2

4

44

2

2

h2

3h

1

3

1

3

(8)

(9)

2

2

4

4

2

2

4

4

6

6

2

2

4

6

(10)

8

10

2k−12k−1

2k−1

2k−1

2k−1

(11)

## For example, student 2 = M C has the mobil phone number of students 4 = SC, 13 = M D, 16 = DA and 21 = II.

AC

AC

M C

M C

CP

CP

SC

SC

SM

SM

RP

RP

M F

M F

AF

AF

F I

F I

DL

DL

EL

EL

CM

CM

M D

M D

GS

GS

SB

SB

DA

DA

J D

J D

AC

AC

GL

GL

M R

M R

I I

I I

(12)

### AC

16 16 16 16 16 16

14 14 14 14

12 12

13 13 13

14 14 14 14

14 14 14 14

12 12

12 12

14 14 14 14

13 13 13

14 14 14 14

14 14 14 14

### JD

16 16 16 16 16 16

14 14 14 14

### GL

17 17 17 17 17 17 17

13 13 13

15 15 15 15 15

(13)

2k−1

1 2

j

j

1

2

j+1

j

j

12

j+1

j

j

2k−j−1

2k−1

j

2k−1

j

2k−1

2k−1

## That is, (CP) is true also for j = 2k.

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