# Metoda Backtracking

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Se adreseaza elevilor de clasa a XI, profil matematica informatica, intensiv informatica sau pentru pregatirea examenului de bacalaureat, disciplina informatica

• 1.

### Un algoritm de tip backtracking genereaza in ordine lexicografica, toate sirurile de 5 cifre 0 si 1 cu proprietatea ca nu exista mai mult de doua cifre de 0 consecutive. Primele sase solutii generate sunt: 00100, 00101, 00110, 00111, 01001, 01010. Care este cea de-a opta solutie?

• A.

01110

• B.

01100

• C.

01011

• D.

01101

B. 01100
Explanation
The given algorithm is generating all the 5-digit binary strings in lexicographic order, with the condition that there are no more than two consecutive zeros. The first six solutions generated are 00100, 00101, 00110, 00111, 01001, and 01010. The eighth solution can be determined by continuing the lexicographic order, so the next string after 01010 is 01100. Therefore, the eighth solution is 01100.

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• 2.

### Un algoritm backtracking genereaza toate sirurile alcatuite din cate 6 cifre binare (0 si 1). Numarul tuturor solutiilor generate va fi egal cu :

• A.

64

• B.

32

• C.

16

• D.

12

A. 64
Explanation
The algorithm generates all possible combinations of 6 binary digits (0 and 1). Since there are 2 options for each digit, the total number of solutions generated will be equal to 2^6, which is 64.

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• 3.

### Aplicand metoda backtracking pentru a genera toate permutarile celor n elemente ale unei multimi, o solutie se memoreaza sub forma unui tablou unidimensional x1, x2, ..., xn. Daca sunt deja generate valori pentru componentele x1, x2, ..., xk-1, iar pentru componenta xk (1 <k<n) au fost testate toate valorile posibile si nu a fost gasita niciuna convenabila, atunci:

• A.

Se incearca alegerea unei valori pentru componenta xk-1.

• B.

se incearca alegerea unei noi valori pentru componenta x1, oricare ar fi valoarea k.

• C.

se incheie algoritmul.

• D.

Se incearca alegerea unei valori pentru componenta xk+1.

A. Se incearca alegerea unei valori pentru componenta xk-1.
Explanation
If all possible values for the component xk have been tested and none of them are suitable, the algorithm tries to choose a value for the previous component xk-1. This suggests that the algorithm is backtracking and attempting to find a different value for a previous component in order to continue generating permutations.

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• 4.

### Daca se utilizeaza metoda backtracking pentru a genera toate numerele naturale, in ordine strict crescatoare, formate din 4 cifre pare distincte, care dintre numerele de mai jos trebuie, eliminate astfel incat cele ramase sa reprezinte o succesiune de numere corect generate? 1) 2068; 2) 2084; 3) 2088; 4) 2468; 5) 2086; 6) 2406

• A.

numai 3)

• B.

atat 3) cat si 5)

• C.

Atat 3) cat si 4)

• D.

numai 4)

C. Atat 3) cat si 4)
Explanation
The numbers that need to be eliminated are 3) 2088 and 4) 2468. In order to generate a sequence of distinct even 4-digit numbers in strictly increasing order using backtracking, we need to ensure that each number in the sequence is formed by adding a digit to the previous number. However, both 2088 and 2468 violate this condition. 2088 is not strictly greater than the previous number 2084, and 2468 does not have distinct digits compared to the previous number 2084. Therefore, both 3) and 4) need to be eliminated.

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• 5.

### Se considera multimea {1, 7, 5, 16, 12}. Se genereaza prin metoda backtracking toate submultimile sale formate din exact 3 elemente: primele patru solutii generate sunt, in ordine: {1, 7, 5}, {1, 7, 16}, {1, 7, 12}. Care dintre solutii trebuie eliminate din sirul urmator astfel incat cele ramase sa apara in sir in ordinea generarii lor: {1, 16, 12}, {5, 16, 12}, {7, 5, 16}, {7, 5, 12}

• A.

{1, 16, 12}

• B.

{5, 16, 12}

• C.

{7, 5, 16}

• D.

{7, 5, 12}

B. {5, 16, 12}
Explanation
The solution {5, 16, 12} should be eliminated from the given sequence. This is because the question asks for the solutions to appear in the order they were generated. The first four solutions generated were {1, 7, 5}, {1, 7, 16}, {1, 7, 12}, and {5, 16, 12}. Therefore, the solution {5, 16, 12} should be eliminated as it was the fourth solution generated and does not appear in the given sequence in the order it was generated.

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• 6.

### Se considera algoritmul care genereaza in ordine strict crescatoare toate numerele formate cu 5 cifre distincte alese din multimea {1, 0, 5, 7, 9} in care cifra din mijloc este 0.Selectati numarul care precede si numarul care urmeaza secventei de numere generate: 19075; 51079; 51097

• A.

19057, 57019

• B.

15079, 71059

• C.

19057, 59071

• D.

15097, 71095

A. 19057, 57019
Explanation
The algorithm generates all numbers formed with 5 distinct digits chosen from the set {1, 0, 5, 7, 9} in strictly increasing order, with the middle digit being 0. The numbers before and after the given sequence are 19057 and 57019, respectively.

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• 7.

### Daca pentru generarea tuturor submultimilor unei multimi A = {1, 2, ..., n} cu 1 â‰¤ n â‰¤ 10, se utilizeaza un algoritm backtracking astfel incat se afiseaza in ordine, pentru n=3, submultimile {}, {1}, {2}, {3},{1, 2}, {1,3}, {2,3}, {1, 2, 3}, atunci, utilizand exact acelasi algoritm pentr n = 4, in sirul submultimilor generate, solutia a 7-a va fi:

• A.

{1,3}

• B.

{4}

• C.

{1,2,3}

• D.

{1,4}

A. {1,3}
Explanation
The given answer is {1,3} because it is the seventh subset generated in the sequence of subsets for n=4 using the backtracking algorithm. The algorithm generates subsets in a specific order, and {1,3} is the seventh subset in that order.

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• 8.

### Produsul cartezia {1,2,3}x{2,3} este obtinut cu ajutorul unui algoritm backtracking care genereaza perechile (1,2), (1,3), (2,2), (2,3), (3,2) si (3,3). Care este numarul perechilor obtinute prin utilizarea aceluiasi algoritm la generarea produsului cartezian {1, 2, 3, 4, 5}x{a, b, c, d}?

• A.

9

• B.

20

• C.

10

• D.

6

B. 20
Explanation
The algorithm generates pairs by combining each element from the first set with each element from the second set. In this case, the first set has 3 elements and the second set has 4 elements. Therefore, the total number of pairs that can be generated is 3 x 4 = 12. However, the algorithm also generates pairs (3, a), (3, b), (3, c), and (3, d) but these pairs are not included in the given options. Therefore, the correct answer is 12 - 4 = 8.

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• 9.

### Se genereaza toate sirurile strict crescatoare de numere naturale nenule mai mici sau egale cu 4, avand primul termen 1 sau 2, ultimul termen 4 si cu diferenta dintre oricare doi termeni aflati pe pozitii consecutive cel mult 2, obtinandu-se solutiile (1, 2, 3,4), (1, 2, 4), (1, 3, 4), (2, 3, 4), (2, 4). Folosind aceeasi metoda generam toate sirurile strict crescatoare de numere naturale nenule mai mic sau egale cu 6, avand primul termen 1 sau 2, ultimul termen 6 si diferenta dintre oricare doi termeni aflati pe pozitii consecutive cel mult 2, care dintre afirmatiile urmatoare este adevarata:

• A.

Imediat dupa solutia (1, 3, 4, 5, 6) se genereaza solutia (2, 3, 4, 5, 6)

• B.

Penultima solutie generata este (2, 3, 5, 6)

• C.

Imediat dupa solutia (1, 2, 4, 6) se genereaza solutia (1, 3, 4, 6)

• D.

In total sunt generate 13 solutii.

D. In total sunt generate 13 solutii.
Explanation
The explanation for the given correct answer is that the question asks for the total number of solutions generated. By following the given method of generating strictly increasing sequences of natural numbers less than or equal to 6, with the first term being 1 or 2, the last term being 6, and the difference between any two consecutive terms being at most 2, we can find that a total of 13 solutions are generated.

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• 10.

### Avand la dispozitie cifrele 0, 1 si 2 putem genera, in ordine crescatoare, numerele care au suma cifrelor egala cu 2 astfel: 2, 11, 20, 101, 110, 200, etc. Folosind acest algoritm generati numerele cu cifrele 0, 1 si 2 care au suma cifrelor egala cu 3. Care va fi al saptelea numar din aceasta generare?

• A.

120

• B.

1002

• C.

201

• D.

210

D. 210
Explanation
The algorithm described in the question generates numbers with digits 0, 1, and 2 in increasing order, where the sum of the digits is equal to 2. To find the seventh number in this generation, we continue following the algorithm. The numbers generated so far are 2, 11, 20, 101, 110, and 200. The next number in the sequence would be 201. Therefore, the seventh number in this generation is 210.

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• 11.

### Generarea tuturor cuvintelor de 4 litere, fiecare litera putand fi orice element din multimea {a, c, e, m, v, s}, se realizeaza cu ajutorul unui algoritm echivalent cu algoritmul de generare a:

• A.

Produsului cartezian

• B.

Combinarilor

• C.

Partitiilor unei multimi

• D.

Permutarilor

A. Produsului cartezian
Explanation
The algorithm for generating all 4-letter words from the set {a, c, e, m, v, s} is equivalent to the algorithm for generating the Cartesian product. This is because the Cartesian product of two sets is the set of all possible ordered pairs of elements from those sets. In this case, we can consider each letter in the word as an element from the set {a, c, e, m, v, s}, and the 4-letter word as an ordered pair of elements. Therefore, the algorithm for generating all 4-letter words is equivalent to generating the Cartesian product of the set {a, c, e, m, v, s} with itself four times.

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• 12.

### Folosind un algoritm de generare putem obtine numere naturale de k cifre care au suma cifrelor egala cu un numar natural s introdus de la tastatura, unde s si k sunt numere naturale nenule. Astfel pentru valorile k = 2 si s = 6 se genereaza numerele: 15, 24, 33, 42, 51, 60. Care vor fi primele 4 numere ce se vor genera pentru k = 3 si s=8?

• A.

800, 710, 620, 530

• B.

107, 116, 125, 134

• C.

125, 233, 341, 431

• D.

116, 125, 134, 143

B. 107, 116, 125, 134
Explanation
The question asks for the first 4 numbers that will be generated using the given algorithm for k = 3 and s = 8. The algorithm generates numbers with k digits and the sum of the digits equal to s. To find the first 4 numbers, we start with the smallest possible number (in this case, 100) and increment it by 1 until we find 4 numbers that meet the criteria. The first 4 numbers that satisfy the conditions are 107, 116, 125, and 134.

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• 13.

### Se considera multimile A = {1, 2, 3}, B = {1}, C = {2, 3, 4}. Elementele produsului cartezian AxBxC se genereaza, in ordine astfel: (1, 1, 2), (1, 1, 3), (1, 1, 4), (2, 1, 2), (2, 1, 3), (2, 1, 4), (3, 1, 2), (3, 1, 3), (3, 1, 4). Daca prin acelasi algoritm se genereaza produsul cartezian al multimilor AxBxC, unde A ={a, b}, B ={a}, C = {b, c, d}, atunci cel de-al cincilea element generat este:

• A.

(a, a, d)

• B.

(a, a, c)

• C.

(b, a, b)

• D.

(b, a, c)

D. (b, a, c)
Explanation
The given question provides the sets A, B, and C, and asks for the fifth element in the Cartesian product AxBxC. The Cartesian product is generated by taking each element from A, combining it with each element from B, and then combining that result with each element from C. In this case, the fifth element generated is (b, a, c).

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• 14.

### Pentru a determina toate modalitatile de a scrie numarul 8 ca suma de numere naturale nenule distincte (abstractie facand de ordinea termenilor) se foloseste metoda backtracking obtinandu-se, in ordine, toate solutiile 1+2+5, 1+3+4, 1+7, 2+6, 3+5. Aplicand exact acelasi procedeu, se determina solutiile pentru scrierea numarului 10. Cate solutii de forma 1+ ... exista?

• A.

3

• B.

4

• C.

5

• D.

6

C. 5
Explanation
The given explanation states that the method of backtracking is used to determine all the ways to write the number 8 as a sum of distinct positive integers, disregarding the order of the terms. The solutions obtained in order are 1+2+5, 1+3+4, 1+7, 2+6, and 3+5. It further states that applying the same procedure, the solutions for writing the number 10 are determined. The question asks for the number of solutions of the form 1+... The answer is 5, indicating that there are 5 solutions of the form 1+ for the number 10.

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• 15.

### Se considera multimile A = {1, 2, 3}, B = {1}, C = {2, 3, 4}. Elementele produsului cartezian AxBxC se genereaza, folosind metoda backtracking, in ordinea (1, 1, 2), (1, 1, 3), (1, 1, 4), (2, 1, 2), (2, 1, 3), (2, 1, 4), (3, 1, 2), (3, 1, 3), (3, 1, 4). Daca prin acelasi algoritm se genereaza produsul cartezian al multimilor AxBxC unde A = {x, y}, B = {x, u}, c = {x, y, z}, atunci cel de-al saptelea element generat este:

• A.

(y, u, x)

• B.

(y, x, x)

• C.

(y, x, z)

• D.

(y, y, z)

B. (y, x, x)
Explanation
The given question states that the elements of the Cartesian product AxBxC are generated using the backtracking method. The order in which the elements are generated is (1, 1, 2), (1, 1, 3), (1, 1, 4), (2, 1, 2), (2, 1, 3), (2, 1, 4), (3, 1, 2), (3, 1, 3), (3, 1, 4).

In the second scenario, the sets A, B, and C are different. A = {x, y}, B = {x, u}, and C = {x, y, z}. Using the same backtracking algorithm, the seventh element generated would be (y, x, x).

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• 16.

### Generarea tuturor sirurilor formate din trei elemente, fiecare element putand fi oricare numar din multimea {1, 2, 3}, se realizeaza cu ajutorul unui algoritm echivalent cu algoritmul de generare a:

• A.

Permutarilor

• B.

Combinarilor

• C.

Produsului cartezian

• D.

Aranjamentelor

C. Produsului cartezian
Explanation
The question asks for the algorithm used to generate all possible strings of three elements, where each element can be any number from the set {1, 2, 3}. The correct answer is "produsului cartezian" which translates to "Cartesian product" in English. The Cartesian product of two sets is a set of all possible ordered pairs formed by taking one element from each set. In this case, the Cartesian product of the set {1, 2, 3} with itself is used to generate all possible combinations of three elements.

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• 17.

### In utilizarea metodei backtracking pentru a genera toate cuvintele alcatuite din doua litere ale multimii {a, c, e, q}, astfel incat sa nu existe doua consoane alaturate, cuvintele se genereaza in urmatoarea ordine: aa, ac, ae, aq, ca, ce, ea, ec, ee, eq, qa, qe. Daca se utilizeaza exact aceeasi metoda pentru a genera cuvinte formate din 4 litere ale multimii {a, b, c, d, e, f}, astfel incat sa nu existe doua consoane alaturate in cuvant, care este penultimul cuvant generat?

• A.

Fefa

• B.

Fafe

• C.

Feef

• D.

Fefe

A. Fefa
Explanation
The given question states that the words are generated using the backtracking method, where two consonants cannot be adjacent. The words are generated in a specific order. By following the same method for generating words from the set {a, b, c, d, e, f}, the penultimate word generated would be "fefa".

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• 18.

### Utilizand metoda backtracking se genereaza toate numerele formate doar din trei cifre astfel incat fiecare numar sa aiba cifrele distincte. Cifrele fiecarui numar sunt din multimea {12, 2, 3, 4}. acest algoritm genereaza numerele, in aceasta ordine: 123, 124, 132, 134, 213, 214, 231, 234, 312, 314, 321, 324, 412, b413, 421, 423, 431, 432. Daca utilizam acelasi algoritm pentru a genera toate numerele de 4cifre, fiecare numar fiind format din cifre distincte din multimea {1, 2, 3, 4, 5}, precizati care este numarul generat imedia dupa 4325.

• A.

4351

• B.

5123

• C.

4521

• D.

4321

A. 4351
Explanation
The backtracking algorithm generates all numbers formed by three digits with distinct digits from the set {12, 2, 3, 4}. The numbers are generated in the following order: 123, 124, 132, 134, 213, 214, 231, 234, 312, 314, 321, 324, 412, 413, 421, 423, 431, 432. If we use the same algorithm to generate all four-digit numbers with distinct digits from the set {1, 2, 3, 4, 5}, the number immediately after 4325 is 4351.

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• Mar 22, 2023
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• Oct 24, 2010
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