## Description

1 Obtaining a sum from a subsequence of digits

Writeaprogramthatpromptstheuserfortwonumbers,sayavailable_digitsanddesired_sum, and outputs the number of ways of selecting digits from available_digits that sum up to desired_sum. For instance, if available_digits is 12234 and sum is 5 then there are 4 solutions:

• one solution is obtained by selecting 1 and both occurrences of 2 (1+2+2=5); • one solution is obtained by selecting 1 and 4 (1+4=5); • one solution is obtained by selecting the ﬁrst occurrence of 2 and 3 (2+3=5); • one solution is obtained by selecting the second occurrence of 2 and 3 (2+3=5).

Here is a possible interaction:

$ python question_1.py Input a number that we will use as available digits: 12234 Input a number that represents the desired sum: 5 There are 4 solutions. $ python question_1.py Input a number that we will use as available digits: 11111 Input a number that represents the desired sum: 5 There is a unique solution $ python question_1.py Input a number that we will use as available digits: 11111 Input a number that represents the desired sum: 6 There is no solution. $ python question_1.py Input a number that we will use as available digits: 1234321 Input a number that represents the desired sum: 5 There are 10 solutions.

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2 R Merging two strings into a third one

Say that two strings s1 and s2 can be merged into a third string s3 if s3 is obtained from s1 by inserting arbitrarily in s1 the characters in s2, respecting their order. For instance, the two strings ab and cd can be merged into abcd, or cabd, or cdab, or acbd, or acdb, …, but not into adbc nor into cbda. Write a program that prompts the user for 3 strings and displays the output as follows: • If no string can be obtained from the other two by merging, then the program outputs that there is no solution. • Otherwise, the program outputs which of the strings can be obtained from the other two by merging.

Here is a possible interaction:

$ python question_2.py Please input the first string: ab Please input the second string: cd Please input the third string: abcd The third string can be obtained by merging the other two. $ python question_2.py Please input the first string: ab Please input the second string: cdab Please input the third string: cd The second string can be obtained by merging the other two. $ python question_2.py Please input the first string: abcd Please input the second string: cd Please input the third string: ab The first string can be obtained by merging the other two. $ python question_2.py Please input the first string: ab Please input the second string: cd Please input the third string: adcb No solution $ python question_2.py Please input the first string: aaaaa Please input the second string: a Please input the third string: aaaa The first string can be obtained by merging the other two. $ python question_2.py Please input the first string: aaab Please input the second string: abcab Please input the third string: aaabcaabb The third string can be obtained by merging the other two. $ python question_2.py Please input the first string: ??got Please input the second string: ?it?go#t## Please input the third string: it### The second string can be obtained by merging the other two.

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3 The n-queens puzzle

This is a well known puzzle: place n chess queens on an n × n chessboard so that no queen is attacked by any other queen (that is, no two queens are on the same row, or on the same column, or on the same diagonal). There are numerous solutions to this puzzle that illustrate all kinds of programming techniques. You will ﬁnd lots of material, lots of solutions on the web. You can of course start with the wikipedia page: http://en.wikipedia.org/wiki/Eight_queens_puzzle. You should try and solve this puzzle in any way you like. Onesetoftechniqueconsistsingeneratingpermutationsof[0,1,…,n−1],apermutation[k0,k1,…,kn−1] requesting to place the queen of the ﬁrst row in the (k0 +1)st column, the queen of the second row in the (k1 +1)nd column, etc. For instance, with n = 8 (the standard chessboard size), the permutation [4, 6, 1, 5, 2, 0, 3, 7] gives rise to the solution:

0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1

The program cryptarithm_v1.py, written by Raymond Hettinger as part of ActiveState Code Recipes, demonstrates the use of the permutations() function from the itertools modules. With cryptarithm_v2.py, we have a variant with an implementation of Heap’s algorithm to generate permutations and a technique to ’skip’ some of them. We could do the same here. For instance, starting with [0,1,2,3,4,5,6,7], we ﬁnd out that the queen on the penultimate row is attacked by the queen on the last row, and skip all permutations of [0,1,2,3,4,5,6,7] that end in[6,7]. IfyouhaveacquiredagoodunderstandingofthedescriptionofHeap’salgorithmgivenin Permutations.pdf and of cryptarithm_v2.py, then try and solve the n-queens puzzle generating permutations and skipping some using Heap’s algorithm; this is the solution I will provide. Doing so will bring your understanding of recursion to new levels, but it is not an easy problem, only attempt it if you want to challenge yourself…

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Hereisapossibleinteraction. Itisinterestingtoprintoutthenumberofpermutationsbeingtested.

$ python … from question_3 import * puzzle = QueenPuzzle(8) puzzle.print_nb_of_tested_permutations() 3544 puzzle.print_nb_of_solutions() 92 puzzle.print_solution(0) 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 puzzle.print_solution(45) 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 puzzle = QueenPuzzle(11) puzzle.print_nb_of_tested_permutations() 382112 puzzle.print_nb_of_solutions() 2680 puzzle.print_solution(1346) 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0