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Prison Cell math Problem!?
Imagine you have a prison that has 100 cells, all locked.
Suppose the first guard goes through and unlocks every cell,
the second guard goes through and changes the state of every other cell starting with #2
then the 3rd guard goes through and changes the state of every third cell begginging with #3,
The fourth guard changes the state of every 4th locker starting with #4,
Imagine that this continues until 100 guards have followed that pattern with the 100 cells.
Alright so I understand that it has to do with Perfect Squares but im pretty lost,
Questions: 1.) What doors will be left unlocked?
2.) Which doors will hav been locked or unlocked only twice?
3.) Which doors will have been locked or unlocked exactly three times?
4.) Which doors will have been locked or unlocked exactly 4 times?
5.) Which doors will have been locked or unlocked only once?
6.) How can you tell exactly how many times a certain door may have been locked or unlocked?
7.) Which cell has been locked or unlocked the most? How many Times?
Someone help me please!
4 Answers
- BenLv 710 years agoFavorite Answer
1) the prison cells that are left unlocked are the ones which are "changed" an odd number of times. You can tell how many times a cell is changed by looking at its factors.
For example, let's look at cell #10:
10 has the factors 1, 2, 5, and 10. So, it will be changed once by each of the guards 1, 2, 5, and 10. Since it is changed an even total number of times, the cell will be locked.
So, if we're looking for a cell that becomes unlocked, we need a cell with an odd number of factors. As it so happens, the only lockers that can have an odd number of factors are perfect squares. For example, 9 has the factors 9, 3, and 1, which means it will be changed 3 times, leaving it unlocked. From this logic, it's clear that 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 will stay unlocked.
2) For a door to have only been changed twice, it must have exactly two factors. That is the characteristic of a PRIME NUMBER, whose factors are 1 and itself. So, the answer to this will be any primes between 1 and 100
3) Any doors which have exactly 3 factors, which is only the squares of prime numbers. So, that's 4, 9, 25, and 49.
4) Any doors that are the product of two distinct primes
5) Door #1
6) As I've said, by the number of factors
7) Which number has the most factors? This question isn't so easy. I believe the answer will be either 60 or 90, which both have 12 factors and are thus changed 12 times.
- zLv 510 years ago
1.) left unlocked: 1 4 9 16 25 36 49 64 81 100
5.) locked or unlocked only once: 1 2 3 4 5 7 9 11 13 17 19 23 25 29 31 37 41 43 47 49 53 59 61 67 71 73 79 83 89 97
2.) locked or unlocked only twice: 4 6 8 9 10 14 15 16 21 22 25 26 27 33 34 35 38 39 46 49 51 55 57 58 62 65 69 74 77 81 82 85 86 87 91 93 94 95
3.) locked or unlocked exactly 3 times: 12 16 18 20 28 32 44 45 50 52 63 64 68 75 76 81 92 98 99
4.) ocked or unlocked exactly 4 times: 24 30 36 40 42 54 56 64 66 70 78 88 100
7.) max is 6 times. Cells are 60 72 84 90 96
6.) here is the complete solution generated by computer: [L/U: 1=locked, 0=unlocked. #L=No. of times locked, #U=No. of times unlocked]
cell L/U #L #U
1 0 0 1
2 1 1 1
3 1 1 1
4 0 1 2
5 1 1 1
6 1 2 2
7 1 1 1
8 1 2 2
9 0 1 2
10 1 2 2
11 1 1 1
12 1 3 3
13 1 1 1
14 1 2 2
15 1 2 2
16 0 2 3
17 1 1 1
18 1 3 3
19 1 1 1
20 1 3 3
21 1 2 2
22 1 2 2
23 1 1 1
24 1 4 4
25 0 1 2
26 1 2 2
27 1 2 2
28 1 3 3
29 1 1 1
30 1 4 4
31 1 1 1
32 1 3 3
33 1 2 2
34 1 2 2
35 1 2 2
36 0 4 5
37 1 1 1
38 1 2 2
39 1 2 2
40 1 4 4
41 1 1 1
42 1 4 4
43 1 1 1
44 1 3 3
45 1 3 3
46 1 2 2
47 1 1 1
48 1 5 5
49 0 1 2
50 1 3 3
51 1 2 2
52 1 3 3
53 1 1 1
54 1 4 4
55 1 2 2
56 1 4 4
57 1 2 2
58 1 2 2
59 1 1 1
60 1 6 6
61 1 1 1
62 1 2 2
63 1 3 3
64 0 3 4
65 1 2 2
66 1 4 4
67 1 1 1
68 1 3 3
69 1 2 2
70 1 4 4
71 1 1 1
72 1 6 6
73 1 1 1
74 1 2 2
75 1 3 3
76 1 3 3
77 1 2 2
78 1 4 4
79 1 1 1
80 1 5 5
81 0 2 3
82 1 2 2
83 1 1 1
84 1 6 6
85 1 2 2
86 1 2 2
87 1 2 2
88 1 4 4
89 1 1 1
90 1 6 6
91 1 2 2
92 1 3 3
93 1 2 2
94 1 2 2
95 1 2 2
96 1 6 6
97 1 1 1
98 1 3 3
99 1 3 3
100 0 4 5
- Anonymous10 years ago
You are right now in gale surfing the Internet from your iPhone and asking math problems that actually are the escape plan. We won't give you any idea.
