comp9021 – Lab 2 (Solution)

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COMP9021, Session 1, 2015
1 R Perfect numbers
A number is perfect if it is equal to the sum of its divisors, itself excluded. For instance, the divisors of 28 distinct from 28 are 1, 2, 4, 7 and 14, and 1 + 2 + 4 + 7 + 14 = 28, hence 28 is perfect.
Write a program that outputs all 3-digit perfect numbers.
You should find out that here is only one solution, equal to 496.
2 R Finding special triples of the form (n,n +1,n +2)
Write a program that finds all triples of consecutive positive three-digit integers each of which is the sum of two squares, that is, all triples of the form (n,n + 1,n + 2) such that:
โ€ข n, n + 1 and n + 2 are integers at least equal to 100 and at most equal to 999;
โ€ข each of n, n + 1 and n + 2 is of the form a2 + b2.
Hint: As we are not constrained by memory space for this problem, we might use a list that stores an integer for all indexes n in [100,999], equal to 1 in case n is the sum of two squares, and to 0 otherwise. Then it is just a matter of finding three consecutive 1โ€™s in the list. This idea can be refined (by not storing 1s, but suitable nonzero values) to not only know that some number is of the form a2 + b2, but also know such a pair (a,b)…
The output of the program could be:
(144, 145, 146) (equal to (0^2+12^2, 8^2+9^2, 5^2+11^2)) is a solution.
(232, 233, 234) (equal to (6^2+14^2, 8^2+13^2, 3^2+15^2)) is a solution.
(288, 289, 290) (equal to (12^2+12^2, 8^2+15^2, 11^2+13^2)) is a solution.
(360, 361, 362) (equal to (6^2+18^2, 0^2+19^2, 1^2+19^2)) is a solution.
(520, 521, 522) (equal to (14^2+18^2, 11^2+20^2, 9^2+21^2)) is a solution.
(576, 577, 578) (equal to (0^2+24^2, 1^2+24^2, 17^2+17^2)) is a solution.
(584, 585, 586) (equal to (10^2+22^2, 12^2+21^2, 15^2+19^2)) is a solution.
(800, 801, 802) (equal to (20^2+20^2, 15^2+24^2, 19^2+21^2)) is a solution.
(808, 809, 810) (equal to (18^2+22^2, 5^2+28^2, 9^2+27^2)) is a solution.
3 Estimating the probabilities of hypotheses in the light of more and more evidence
Write a program that simulates the cast of an unknown die, chosen from a set of 5 dice with 4, 6, 8, 12, and 20 faces. To start with, every die has a probability of 0.2 to be the chosen die. At every cast, the probability of each die being the chosen die is updated using Bayesโ€™ rule (find out about it if you do not know it, it is simple…). The probabilities are displayed for at most 5 casts. If more than 5 casts have been requested, the final probabilities obtained for the chosen number of casts are eventually displayed.
Here is a possible output:
Enter the desired number of times a randomly chosen die will be cast: 2
This is a secret, but the chosen die is the one with 4 faces
The die that has been cast yields: 4
The updated dice probabilies are:
4: 37.04%
6: 24.69%
8: 18.52%
12: 12.35%
20: 7.41%
The die that has been cast yields: 1 The updated dice probabilies are:
4: 54.18%
6: 24.08%
8: 13.55%
12: 6.02%
20: 2.17%
Here is another possible output:
Enter the desired number of times a randomly chosen die will be cast: 5
This is a secret, but the chosen die is the one with 8 faces
The die that has been cast yields: 7
The updated dice probabilies are:
4: 0.00%
6: 0.00%
8: 48.39%
12: 32.26%
20: 19.35%
The die that has been cast yields: 1 The updated dice probabilies are:
4: 0.00%
6: 0.00%
8: 62.33%
12: 27.70%
20: 9.97%
The die that has been cast yields: 1 The updated dice probabilies are:
4: 0.00%
6: 0.00%
8: 73.51%
12: 21.78%
20: 4.70%
The die that has been cast yields: 3 The updated dice probabilies are:
4: 0.00%
6: 0.00%
8: 81.76%
12: 16.15%
20: 2.09%
The die that has been cast yields: 7 The updated dice probabilies are:
4: 0.00%
6: 0.00%
8: 87.57%
12: 11.53%
20: 0.90%
Here is still another possible output:
Enter the desired number of times a randomly chosen die will be cast: 20
This is a secret, but the chosen die is the one with 6 faces
The die that has been cast yields: 1 The updated dice probabilies are:
4: 37.04%
6: 24.69%
8: 18.52%
12: 12.35%
20: 7.41%
The die that has been cast yields: 4 The updated dice probabilies are:
4: 54.18%
6: 24.08%
8: 13.55%
12: 6.02%
20: 2.17%
The die that has been cast yields: 6 The updated dice probabilies are:
4: 0.00%
6: 63.54%
8: 26.80%
12: 7.94%
20: 1.72%
The die that has been cast yields: 6 The updated dice probabilies are:
4: 0.00%
6: 72.10%
8: 22.81%
12: 4.51%
20: 0.58%
The die that has been cast yields: 5 The updated dice probabilies are:
4: 0.00%
6: 78.68%
8: 18.67%
12: 2.46%
20: 0.19%
The final probabilities are:
4: 0.00%
6: 99.68%
8: 0.32%
12: 0.00%
20: 0.00%

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