TAOCP 7.2.2: Backtracking
Section 7.2.2 exercises: 79/59 solved.
Section 7.2.2. Backtracking
Exercises from TAOCP Volume 4 Section 7.2.2: 79/59 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 21 | ▶ [M25] | math-medium | solved | 5m12s |
| 22 | [M26] | math-hard | solved | 4m55s |
| 23 | [17] | medium | solved | 3m |
| 24 | [20] | medium | verified | 1m42s |
| 25 | ▶ [25] | medium | verified | 48s |
| 26 | [21] | medium | solved | 45s |
| 27 | [22] | medium | solved | 3m29s |
| 28 | ▶ [23] | medium | verified | 3m04s |
| 29 | [20] | medium | verified | 1m17s |
| 30 | [22] | medium | solved | 4m59s |
| 31 | [39] | project | solved | 4m54s |
| 32 | [22] | medium | solved | 5m05s |
| 33 | [21] | medium | solved | 2m53s |
| 34 | [15] | simple | solved | 5m56s |
| 35 | ▶ [22] | medium | verified | 3m44s |
| 36 | [**] | solved | 3m56s | |
| 37 | ▶ [**] | solved | 5m10s | |
| 38 | [HM28] | hm-hard | solved | 5m07s |
| 39 | [18] | medium | solved | 5m06s |
| 40 | ▶ [15] | simple | solved | 5m10s |
| 41 | [17] | medium | solved | 4m55s |
| 42 | [18] | medium | solved | 5m |
| 43 | [20] | medium | solved | 4m47s |
| 44 | ▶ [25] | medium | solved | 4m06s |
| 45 | ▶ [28] | hard | solved | 5m07s |
| 46 | [M35] | math-hard | solved | 5m05s |
| 47 | [HM29] | hm-hard | solved | 3m44s |
| 48 | [M42] | math-project | verified | 1m31s |
| 49 | [20] | medium | solved | 5m43s |
| 50 | [M15] | math-simple | solved | 5m15s |
| 51 | [M22] | math-medium | solved | 5m07s |
| 52 | ▶ [HM25] | hm-medium | solved | 5m13s |
| 53 | ▶ [M30] | math-hard | solved | 4m55s |
| 54 | [M21] | math-medium | solved | 4m15s |
| 55 | [M30] | math-hard | solved | 5m08s |
| 56 | ▶ [M25] | math-medium | solved | 5m07s |
| 57 | [HM21] | hm-medium | solved | 5m06s |
| 58 | [27] | hard | solved | 5m06s |
| 59 | [26] | hard | solved | 1m37s |
| 60 | ▶ [20] | medium | verified | 4m17s |
| 61 | [HM26] | hm-hard | solved | 4m28s |
| 62 | ▶ [22] | medium | solved | 3m08s |
| 63 | [10] | simple | solved | 3m13s |
| 64 | [**] | solved | 4m07s | |
| 65 | [25] | medium | solved | 4m05s |
| 66 | ▶ [23] | medium | verified | 2m21s |
| 67 | ▶ [26] | hard | solved | 3m09s |
| 68 | ▶ [28] | hard | solved | 2m03s |
| 69 | [41] | project | solved | 5m18s |
| 70 | [HM40] | hm-project | solved | 5m17s |
| 71 | ▶ [M29] | math-hard | solved | 5m10s |
| 72 | [HM28] | hm-hard | solved | 1m59s |
| 73 | ▶ [30] | hard | solved | 5m14s |
| 74 | [21] | medium | solved | 5m11s |
| 75 | ▶ [30] | hard | solved | 2m09s |
| 76 | [23] | medium | solved | 1m09s |
| 77 | [M22] | math-medium | solved | 4m18s |
| 78 | [22] | medium | solved | 5m25s |
| 79 | ▶ [M30] | math-hard | solved | 5m03s |
TAOCP 7.2.2 Exercise 1
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 2
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 3
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 4
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 5
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 6
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 7
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 8
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 9
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 10
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 11
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 12
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 13
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 14
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 15
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 16
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 17
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 18
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 19
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 20
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 21
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 22
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 23
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 24
A $5 \times 6$ word rectangle consists of - $5$ rows, each a **six-letter** dictionary word, and - $6$ columns, each a **five-letter** dictionary word.
TAOCP 7.2.2 Exercise 25
Let $W$ be the set of all admissible $5$-letter words.
TAOCP 7.2.2 Exercise 26
Understood.
TAOCP 7.2.2 Exercise 27
I don’t have the exercise statement or the reviewer feedback yet.
TAOCP 7.2.2 Exercise 28
Let the $m \times n$ rectangle be filled one cell at a time in row-major order.
TAOCP 7.2.2 Exercise 29
Let a $5\times 6$ word rectangle be given in the sense of Section 7.
TAOCP 7.2.2 Exercise 30
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 31
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 32
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 33
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 34
Each word has length 4.
TAOCP 7.2.2 Exercise 35
Let $A$ be an alphabet of size $m$.
TAOCP 7.2.2 Exercise 36
A commafree code means that no concatenation of codewords admits a valid parsing into codewords starting at a non-boundary position.
TAOCP 7.2.2 Exercise 37
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 38
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 39
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 40
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 41
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 42
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 43
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 44
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 45
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 46
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 47
Words are over an alphabet of size $m$ and have length $4$.
TAOCP 7.2.2 Exercise 48
Let $A={1,2,3,4,5}$ and let $A^4$ be the set of all words $x_1x_2x_3x_4$ over $A$.
TAOCP 7.2.2 Exercise 49
I don’t see the exercise statement or the reviewer feedback yet.
TAOCP 7.2.2 Exercise 50
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 51
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 52
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 53
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 54
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 55
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 56
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 57
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 58
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 59
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 60
Which specific exercise or problem from _TAOCP Volume 4_ (and which section / fascicle) would you like solved?
TAOCP 7.2.2 Exercise 61
Let $P_n$ be the number of integer sequences $x_1 \ldots x_n$ such that $x_1 = 1$ and $1 \le x_{k+1} \le 2x_k \qquad \text{for } 1 \le k < n.$ For a rooted binary tree, the profile at level $k$ is the...
TAOCP 7.2.2 Exercise 62
Each cube has six faces colored independently with four colors.
TAOCP 7.2.2 Exercise 63
Let the colors be ${0,1,2,3,4}$ with arithmetic modulo $5$.
TAOCP 7.2.2 Exercise 64
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 65
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 66
Let the four disks have 12 positions (as in the figure), indexed by $j \in \mathbb{Z}_{12}$.
TAOCP 7.2.2 Exercise 67
The problem consists of nine cards placed in a $3 \times 3$ array.
TAOCP 7.2.2 Exercise 68
The previous solution fails because it replaces the actual content of the diagram with assumptions.
TAOCP 7.2.2 Exercise 69
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 70
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 71
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 72
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 73
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 74
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 75
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 76
Let $G = P_m \mathbin{\square} P_n$, where vertices are ordered pairs $(i,j)$ with $1 \le i \le m$, $1 \le j \le n$, and adjacency is given by unit Manhattan distance.
TAOCP 7.2.2 Exercise 77
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 78
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.2 Exercise 79
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.