Contents

Tutorial 1

Question 1

A time t in seconds can be converted to days, hours, minutes and seconds using integer division:

  1. Divide t by 60; set t to the quotient and call the remainder s.

  2. Divide t by 60; set t to the quotient and call the remainder m.

  3. Divide t by 24; set d to the quotient and call the remainder h.

Calculate the number of days, hours, minutes and seconds in one million seconds. Display the result as follows:

1000000 seconds is xx days H:M:S

Question 2

Iodine-131 is a radioisotope with a half-life \(t_\mathrm{half} = 8.02\) days.

  1. Use the equation \(r = \left(\frac{1}{2}\right) ^ {1/t_\mathrm{half}}\) to calculate the daily decay rate (i.e. the proportion of a sample of Iodine 131 that decays in one day).

  2. A sample of radioactive material containing Iodine-131 has an activity of \(1.67 \times 10^{12}\) Becquerels. Write a program which calculates the number of days it takes for the sample to reach an activity of \(10000\) Becquerels.

  3. Print the activity of the sample once every 7 days.

Question 3

We can calculate how many digits there are in a number by repeatedly dividing by 10 until the number is less than one.

  1. Write peudo-code for this calculation.

  2. Write a Python program to implement it, displaying the result in the format 1234 has 4 digits.

  3. Test your program with a variety of numbers.

  4. Extend your program so that it works for negative numbers.

Question 4 (The Collatz Problem)

Given a positive integer n, the Collatz Operation is defined as follows:

  • If the number is even, divide it by two

  • If the number is odd, triple it and add one

Repeatedly applying the Collatz Operation results in a sequence of integers which eventually reaches the number 1.

Write a program which generates the Collatz Sequence for a given positive integer \(n\).

For example, if \(n = 5\) then your program should produce the following:

5
16
8
4
2
1

Given a positive integer \(n\), define the Collatz Number to be the length of the Collatz Sequence for \(n\). For example, the Collatz number of 5 is 6.

Write a program which calculates and prints the Collatz number of a given number \(n\). For example, if \(n=5\) your program should print The Collatz number of 5 is 6.

Write a program which finds the smallest number \(n\) whose Collatz Number is greater than 200.