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Traveling professor

Question

There is an equilateral pentacontagon (a polygon with 50 sides). In one of its vertex stands Dr. Faust. He has three options 1) walk to the diametrically opposed point free of charge; 2) walk counterclockwise to the neighboring vertex by paying $1.05 to Mephistopheles; 3) walk clockwise to the neighboring vertex by receiving a payment of $1.05 from Mephistopheles. If it is given that Dr. Faust has been everywhere at least once, prove that at some point someone paid no less than $25.

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I can move the good Dr. for less than $25

Spoiler

In 324 total steps the Dr. can visit every vertex for no more than $12.60 going either way.

Method: From the starting point move in one direction (say Clockwise), then move across, then move in the other direction (CCW) until moving across would bring you to a new location. Then move across and begin moving in the original direction until you reach a new vertex. Repeat until completed.

Data. Assume starting at #1 and all vertices are number around clockwise from here.

Spoiler
Start Direction End Cost Running total
1 CW 2  $       1.05  $       1.05
2 A 27  $           -    $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 A 50  $           -    $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 A 28  $           -    $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 A 49  $           -    $    (2.10)
49 CW 50  $       1.05  $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 CW 4  $       1.05  $       3.15
4 A 29  $           -    $       3.15
29 CCW 28  $    (1.05)  $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 CCW 23  $    (1.05)  $    (3.15)
23 A 48  $           -    $    (3.15)
48 CW 49  $       1.05  $    (2.10)
49 CW 50  $       1.05  $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 CW 4  $       1.05  $       3.15
4 CW 5  $       1.05  $       4.20
5 A 30  $           -    $       4.20
30 CCW 29  $    (1.05)  $       3.15
29 CCW 28  $    (1.05)  $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 CCW 23  $    (1.05)  $    (3.15)
23 CCW 22  $    (1.05)  $    (4.20)
22 A 47  $           -    $    (4.20)
47 CW 48  $       1.05  $    (3.15)
48 CW 49  $       1.05  $    (2.10)
49 CW 50  $       1.05  $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 CW 4  $       1.05  $       3.15
4 CW 5  $       1.05  $       4.20
5 CW 6  $       1.05  $       5.25
6 A 31  $           -    $       5.25
31 CCW 30  $    (1.05)  $       4.20
30 CCW 29  $    (1.05)  $       3.15
29 CCW 28  $    (1.05)  $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 CCW 23  $    (1.05)  $    (3.15)
23 CCW 22  $    (1.05)  $    (4.20)
22 CCW 21  $    (1.05)  $    (5.25)
21 A 46  $           -    $    (5.25)
46 CW 47  $       1.05  $    (4.20)
47 CW 48  $       1.05  $    (3.15)
48 CW 49  $       1.05  $    (2.10)
49 CW 50  $       1.05  $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 CW 4  $       1.05  $       3.15
4 CW 5  $       1.05  $       4.20
5 CW 6  $       1.05  $       5.25
6 CW 7  $       1.05  $       6.30
7 A 32  $           -    $       6.30
32 CCW 31  $    (1.05)  $       5.25
31 CCW 30  $    (1.05)  $       4.20
30 CCW 29  $    (1.05)  $       3.15
29 CCW 28  $    (1.05)  $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 CCW 23  $    (1.05)  $    (3.15)
23 CCW 22  $    (1.05)  $    (4.20)
22 CCW 21  $    (1.05)  $    (5.25)
21 CCW 20  $    (1.05)  $    (6.30)
20 A 45  $           -    $    (6.30)
45 CW 46  $       1.05  $    (5.25)
46 CW 47  $       1.05  $    (4.20)
47 CW 48  $       1.05  $    (3.15)
48 CW 49  $       1.05  $    (2.10)
49 CW 50  $       1.05  $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 CW 4  $       1.05  $       3.15
4 CW 5  $       1.05  $       4.20
5 CW 6  $       1.05  $       5.25
6 CW 7  $       1.05  $       6.30
7 CW 8  $       1.05  $       7.35
8 A 33  $           -    $       7.35
33 CCW 32  $    (1.05)  $       6.30
32 CCW 31  $    (1.05)  $       5.25
31 CCW 30  $    (1.05)  $       4.20
30 CCW 29  $    (1.05)  $       3.15
29 CCW 28  $    (1.05)  $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 CCW 23  $    (1.05)  $    (3.15)
23 CCW 22  $    (1.05)  $    (4.20)
22 CCW 21  $    (1.05)  $    (5.25)
21 CCW 20  $    (1.05)  $    (6.30)
20 CCW 19  $    (1.05)  $    (7.35)
19 A 44  $           -    $    (7.35)
44 CW 45  $       1.05  $    (6.30)
45 CW 46  $       1.05  $    (5.25)
46 CW 47  $       1.05  $    (4.20)
47 CW 48  $       1.05  $    (3.15)
48 CW 49  $       1.05  $    (2.10)
49 CW 50  $       1.05  $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 CW 4  $       1.05  $       3.15
4 CW 5  $       1.05  $       4.20
5 CW 6  $       1.05  $       5.25
6 CW 7  $       1.05  $       6.30
7 CW 8  $       1.05  $       7.35
8 CW 9  $       1.05  $       8.40
9 A 34  $           -    $       8.40
34 CCW 33  $    (1.05)  $       7.35
33 CCW 32  $    (1.05)  $       6.30
32 CCW 31  $    (1.05)  $       5.25
31 CCW 30  $    (1.05)  $       4.20
30 CCW 29  $    (1.05)  $       3.15
29 CCW 28  $    (1.05)  $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 CCW 23  $    (1.05)  $    (3.15)
23 CCW 22  $    (1.05)  $    (4.20)
22 CCW 21  $    (1.05)  $    (5.25)
21 CCW 20  $    (1.05)  $    (6.30)
20 CCW 19  $    (1.05)  $    (7.35)
19 CCW 18  $    (1.05)  $    (8.40)
18 A 43  $           -    $    (8.40)
43 CW 44  $       1.05  $    (7.35)
44 CW 45  $       1.05  $    (6.30)
45 CW 46  $       1.05  $    (5.25)
46 CW 47  $       1.05  $    (4.20)
47 CW 48  $       1.05  $    (3.15)
48 CW 49  $       1.05  $    (2.10)
49 CW 50  $       1.05  $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 CW 4  $       1.05  $       3.15
4 CW 5  $       1.05  $       4.20
5 CW 6  $       1.05  $       5.25
6 CW 7  $       1.05  $       6.30
7 CW 8  $       1.05  $       7.35
8 CW 9  $       1.05  $       8.40
9 CW 10  $       1.05  $       9.45
10 A 35  $           -    $       9.45
35 CCW 34  $    (1.05)  $       8.40
34 CCW 33  $    (1.05)  $       7.35
33 CCW 32  $    (1.05)  $       6.30
32 CCW 31  $    (1.05)  $       5.25
31 CCW 30  $    (1.05)  $       4.20
30 CCW 29  $    (1.05)  $       3.15
29 CCW 28  $    (1.05)  $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 CCW 23  $    (1.05)  $    (3.15)
23 CCW 22  $    (1.05)  $    (4.20)
22 CCW 21  $    (1.05)  $    (5.25)
21 CCW 20  $    (1.05)  $    (6.30)
20 CCW 19  $    (1.05)  $    (7.35)
19 CCW 18  $    (1.05)  $    (8.40)
18 CCW 17  $    (1.05)  $    (9.45)
17 A 42  $           -    $    (9.45)
42 CW 43  $       1.05  $    (8.40)
43 CW 44  $       1.05  $    (7.35)
44 CW 45  $       1.05  $    (6.30)
45 CW 46  $       1.05  $    (5.25)
46 CW 47  $       1.05  $    (4.20)
47 CW 48  $       1.05  $    (3.15)
48 CW 49  $       1.05  $    (2.10)
49 CW 50  $       1.05  $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 CW 4  $       1.05  $       3.15
4 CW 5  $       1.05  $       4.20
5 CW 6  $       1.05  $       5.25
6 CW 7  $       1.05  $       6.30
7 CW 8  $       1.05  $       7.35
8 CW 9  $       1.05  $       8.40
9 CW 10  $       1.05  $       9.45
10 CW 11  $       1.05  $    10.50
11 A 36  $           -    $    10.50
36 CCW 35  $    (1.05)  $       9.45
35 CCW 34  $    (1.05)  $       8.40
34 CCW 33  $    (1.05)  $       7.35
33 CCW 32  $    (1.05)  $       6.30
32 CCW 31  $    (1.05)  $       5.25
31 CCW 30  $    (1.05)  $       4.20
30 CCW 29  $    (1.05)  $       3.15
29 CCW 28  $    (1.05)  $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 CCW 23  $    (1.05)  $    (3.15)
23 CCW 22  $    (1.05)  $    (4.20)
22 CCW 21  $    (1.05)  $    (5.25)
21 CCW 20  $    (1.05)  $    (6.30)
20 CCW 19  $    (1.05)  $    (7.35)
19 CCW 18  $    (1.05)  $    (8.40)
18 CCW 17  $    (1.05)  $    (9.45)
17 CCW 16  $    (1.05)  $  (10.50)
16 A 41  $           -    $  (10.50)
41 CW 42  $       1.05  $    (9.45)
42 CW 43  $       1.05  $    (8.40)
43 CW 44  $       1.05  $    (7.35)
44 CW 45  $       1.05  $    (6.30)
45 CW 46  $       1.05  $    (5.25)
46 CW 47  $       1.05  $    (4.20)
47 CW 48  $       1.05  $    (3.15)
48 CW 49  $       1.05  $    (2.10)
49 CW 50  $       1.05  $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 CW 4  $       1.05  $       3.15
4 CW 5  $       1.05  $       4.20
5 CW 6  $       1.05  $       5.25
6 CW 7  $       1.05  $       6.30
7 CW 8  $       1.05  $       7.35
8 CW 9  $       1.05  $       8.40
9 CW 10  $       1.05  $       9.45
10 CW 11  $       1.05  $    10.50
11 CW 12  $       1.05  $    11.55
12 A 37  $           -    $    11.55
37 CCW 36  $    (1.05)  $    10.50
36 CCW 35  $    (1.05)  $       9.45
35 CCW 34  $    (1.05)  $       8.40
34 CCW 33  $    (1.05)  $       7.35
33 CCW 32  $    (1.05)  $       6.30
32 CCW 31  $    (1.05)  $       5.25
31 CCW 30  $    (1.05)  $       4.20
30 CCW 29  $    (1.05)  $       3.15
29 CCW 28  $    (1.05)  $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 CCW 23  $    (1.05)  $    (3.15)
23 CCW 22  $    (1.05)  $    (4.20)
22 CCW 21  $    (1.05)  $    (5.25)
21 CCW 20  $    (1.05)  $    (6.30)
20 CCW 19  $    (1.05)  $    (7.35)
19 CCW 18  $    (1.05)  $    (8.40)
18 CCW 17  $    (1.05)  $    (9.45)
17 CCW 16  $    (1.05)  $  (10.50)
16 CCW 15  $    (1.05)  $  (11.55)
15 A 40  $           -    $  (11.55)
40 CW 41  $       1.05  $  (10.50)
41 CW 42  $       1.05  $    (9.45)
42 CW 43  $       1.05  $    (8.40)
43 CW 44  $       1.05  $    (7.35)
44 CW 45  $       1.05  $    (6.30)
45 CW 46  $       1.05  $    (5.25)
46 CW 47  $       1.05  $    (4.20)
47 CW 48  $       1.05  $    (3.15)
48 CW 49  $       1.05  $    (2.10)
49 CW 50  $       1.05  $    (1.05)
50 CW 1  $       1.05  $           -  
1 CW 2  $       1.05  $       1.05
2 CW 3  $       1.05  $       2.10
3 CW 4  $       1.05  $       3.15
4 CW 5  $       1.05  $       4.20
5 CW 6  $       1.05  $       5.25
6 CW 7  $       1.05  $       6.30
7 CW 8  $       1.05  $       7.35
8 CW 9  $       1.05  $       8.40
9 CW 10  $       1.05  $       9.45
10 CW 11  $       1.05  $    10.50
11 CW 12  $       1.05  $    11.55
12 CW 13  $       1.05  $    12.60
13 A 38  $           -    $    12.60
38 CCW 37  $    (1.05)  $    11.55
37 CCW 36  $    (1.05)  $    10.50
36 CCW 35  $    (1.05)  $       9.45
35 CCW 34  $    (1.05)  $       8.40
34 CCW 33  $    (1.05)  $       7.35
33 CCW 32  $    (1.05)  $       6.30
32 CCW 31  $    (1.05)  $       5.25
31 CCW 30  $    (1.05)  $       4.20
30 CCW 29  $    (1.05)  $       3.15
29 CCW 28  $    (1.05)  $       2.10
28 CCW 27  $    (1.05)  $       1.05
27 CCW 26  $    (1.05)  $           -  
26 CCW 25  $    (1.05)  $    (1.05)
25 CCW 24  $    (1.05)  $    (2.10)
24 CCW 23  $    (1.05)  $    (3.15)
23 CCW 22  $    (1.05)  $    (4.20)
22 CCW 21  $    (1.05)  $    (5.25)
21 CCW 20  $    (1.05)  $    (6.30)
20 CCW 19  $    (1.05)  $    (7.35)
19 CCW 18  $    (1.05)  $    (8.40)
18 CCW 17  $    (1.05)  $    (9.45)
17 CCW 16  $    (1.05)  $  (10.50)
16 CCW 15  $    (1.05)  $  (11.55)
15 CCW 14  $    (1.05)  $  (12.60)
14 A 39  $           -    $  (12.60)
39  

 

       

 

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That is a nifty solution, Tojo! After reading yours, I realized there is a shorter route...

Spoiler

From starting point, move 12 clockwise and then 12 counterclockwise to return to starting point. Then 12 more ccw and 12 cw to return to starting point. That covers 25 points with the max spent/received never being more than 12.60. Move across for free and repeat the moves for the other 25 points.

I noticed that your last set of moves covered so many points that one could skip all the prior "swings" and go straight to the final swings.

 

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The way I understood the question, we would be counting the sum of payments made rather than the net difference. In that case, any complete circuit route of 49 moves does require that at least one of them pays a total of at least $25.20.

It is possible, in 49 moves, that each pays a total of $25.20 for a net of zero with neither exceeding a net of $12.60.

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