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AndrewTrieu
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<!doctype html>
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<html>
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<head>
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<title>Week 8 Programming Assignments (8 points)</title>
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<meta charset="utf-8">
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</head>
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<body>
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<h4><strong>Assignment 8.1: Jumps</strong> (4 points)</h4>
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<div>You are playing a game which has \(n\) levels. You start from the level \(0\) and your goal is to reach the level
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\(n\) by jumping from a level to another. From every level you're able to jump only to \(a\) or \(b\)
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levels higher. In how many different ways can you complete the game?
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</div>
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<br>
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<div>e.g.: let \(n=8\), \(a=2\) and \(b=3\), there are \(4\) different ways to pass the game.</div>
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<ul>
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<ul>
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<li>\(0\rightarrow 2\rightarrow 4\rightarrow 6\rightarrow 8\)</li>
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<li>\(0\rightarrow 2\rightarrow 5\rightarrow 8\)</li>
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<li>\(0\rightarrow 3\rightarrow 5\rightarrow 8\)</li>
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<li>\(0\rightarrow 3\rightarrow 6\rightarrow 8\)</li>
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</ul>
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</ul>
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<div>Create a function <strong>jumps(n: int, a: int, b: int)</strong> in Python which returns the number of all possible ways to complete the game.</div>
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<br>
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<div>Limist (you can assume that):</div>
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<ul>
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<ul>
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<li>\(n \leq 10000\)</li>
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<li>\(1 \leq a < b \leq n\)</li>
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</ul>
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</ul>
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<div>Targets:</div>
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<ol>
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<li>correct solution: \(2\) points</li>
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<li>performs in \(\Theta (n)\) time: \(+2\) points (the solution must be correct)</li>
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</ol>
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<div>A code template with an example program:</div>
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<br>
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<div style="border: 2px solid black">
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<pre>def jumps(n, a, b):
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# TODO
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if __name__ == "__main__":
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print(jumps(4, 1, 2)) # 5
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print(jumps(8, 2, 3)) # 4
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print(jumps(11, 6, 7)) # 0
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print(jumps(30, 3, 5)) # 58
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print(jumps(100, 4, 5)) # 1167937
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</pre>
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</div>
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<br>
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<div>Submit your solution in CodeGrade as <strong>jumps.py</strong>.</div>
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<br><br>
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<br>
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<div>
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<h4><strong>Assignment 8.2: All Sums </strong>(4 points)</h4>
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</div>
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<div>\(A\) is a list consisting of \(n\) integers. How many different sums can be generated with the given integers?</div>
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<br>
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<div>For example:</div>
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<ul>
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<ul>
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<li>list \([1, 2, 3]\) has 6 possible sums: \(1\), \(2\), \(3\), \(4\), \(5\) and \(6\)</li>
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<li>list \([2, 2, 3]\) has 5 possible sums: \(2\), \(3\), \(4\), \(5\) and \(7\)</li>
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</ul>
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</ul>
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<div>Create a function <strong>sums(A: list)</strong> in Python which computes the number of all different sums.</div>
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<br>
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<div>Limits (you can assume that): \(1 \leq n, a_i \leq 100\)</div>
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<br>
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<div>Target: The algorithm computes the awnser in less than \(1\) second (in CodeGrade)</div>
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<ol>
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<li>correct solution: \(2\) points</li>
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<li>performs in \(\Theta (n^3)\) time: \(+2\) points (the solution be must correct)</li>
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<ul>
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<li>(extra: consider the running time of your algorithm)</li>
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</ul>
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</ol>
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<div>A code template with an example program:</div>
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<br>
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<div style="border: 2px solid black">
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<pre>def sums(items):
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# TODO
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if __name__ == "__main__":
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print(sums([1, 2, 3])) # 6
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print(sums([2, 2, 3])) # 5
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print(sums([1, 3, 5, 1, 3, 5])) # 18
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print(sums([1, 15, 5, 23, 100, 55, 2])) # 121
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</pre>
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</div>
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<br>
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<div>Submit your solution in CodeGrade as <strong>sums.py</strong>.</div>
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</body>
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</html>
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def jumps(n, a, b):
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if n < 0:
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return 0
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if n == 0:
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return 1
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# Initialize an array to store the number of ways to reach each position
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num_ways = [0] * (n + 1)
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num_ways[0] = 1
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# Calculate the number of ways to reach each position
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for i in range(n + 1):
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num_ways[i] += num_ways[i - a]
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num_ways[i] += num_ways[i - b]
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return num_ways[n]
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if __name__ == "__main__":
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print(jumps(4, 1, 2)) # 5
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print(jumps(8, 2, 3)) # 4
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print(jumps(11, 6, 7)) # 0
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print(jumps(30, 3, 5)) # 58
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print(jumps(100, 4, 5)) # 1167937
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def sums(arr):
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sums = set()
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for i in range(len(arr)):
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for j in range(i, len(arr)):
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for k in range(j, len(arr)):
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sums.add(arr[i] + arr[j] + arr[k])
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return len(sums)
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# Test
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if __name__ == "__main__":
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print(sums([1, 2, 3])) # 6
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print(sums([2, 2, 3])) # 5
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print(sums([1, 3, 5, 1, 3, 5])) # 18
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print(sums([1, 15, 5, 23, 100, 55, 2])) # 121
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def sums(arr):
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# Create a set to store all the unique sums
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sums = set()
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# Initialize a list with length equal to the length of arr
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# to store the previous results
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prev_results = [0] * len(arr)
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# Iterate through each element in arr
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for i in range(len(arr)):
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# Initialize a temporary set to store the new sums
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temp = set()
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# Iterate through all the previous results
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for j in range(i):
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# Add the current element to each of the previous sums
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# and add the new sum to the set
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temp.add(arr[i] + prev_results[j])
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# Add the current element to the set
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temp.add(arr[i])
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# Update the list of previous results with the new sums
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prev_results[i] = temp
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# Update the set of all sums with the new sums
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sums = sums.union(temp)
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return len(sums)
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# Test
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if __name__ == "__main__":
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print(sums([1, 2, 3])) # 6
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print(sums([2, 2, 3])) # 5
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print(sums([1, 3, 5, 1, 3, 5])) # 18
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print(sums([1, 15, 5, 23, 100, 55, 2])) # 121
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