Traveling professor

BMAD · July 29, 2016

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.

tojo928 · August 3, 2016

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

Reveal hidden contents

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.

Reveal hidden contents

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

nana77 · August 7, 2016

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

Reveal hidden contents

Logophobic · August 8, 2016

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.

Sign In

Traveling professor

Question

BMAD

Link to comment

Share on other sites

3 answers to this question

Recommended Posts

tojo928

Link to comment

Share on other sites

nana77

Link to comment

Share on other sites

Logophobic

Link to comment

Share on other sites

Join the conversation

Recently Browsing 0 members

Browse

Activity