1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
|
#include <iostream>
using namespace std;
int main()
{
int x, y;
cout << "x = ";
cin >> x;
cout << "y = ";
cin >> y;
int n, v = max(abs(x), abs(y)), g = 8 * (v * (v + 1) / 2);
if (y == v || x == v) { // Top row or right column
n = g - x + y - 6 * v;
}
else { // Left column or bottom row
n = g + x - y - 2 * v;
}
printf("n = %d", n);
return 0;
}
/* --------------
* Task condition
* --------------
* You have a spiral of whole numbers, starting from 0, like this:
*
* 36 < 35 < 34 < 33 < 32 < 31 < 30
* V ^
* 37 16 < 15 < 14 < 13 < 12 29
* V V ^ ^
* 38 17 4 < 3 < 2 11 28
* V V V ^ ^ ^
* 39 18 5 0 > 1 10 27 ...
* V V V ^ ^ ^
* 40 19 6 > 7 > 8 > 9 26 51
* V V ^ ^
* 41 20 > 21 > 22 > 23 > 24 > 25 50
* V ^
* 42 > 43 > 44 > 45 > 46 > 47 > 48 > 49
*
* The distance between two neighbouring numbers is 1, meaning 0 is at (0;0), 1 is at (1;0), 38 is at (-3;1).
* You are given the coordinates of a number and you have to find the actual number.
*
* --------------------------
* Explanation of my solution
* --------------------------
*
* My solution is a very mathematical one. First, I divide the spiral into squares,
* depending on the offset of 0;0.
* Example:
* - Square 1 (because x and y are between -1 and 1 (one of them is always 1 or -1)):
* 4 < 3 < 2
* V ^
* 5 1
* V
* 6 > 7 > 8
*
* - Square 2 (because x and y are between -2 and 2 (one of them is always 2 or -2)):
* 16 < 15 < 14 < 13 < 12
* V ^
* 17 11
* V ^
* 18 10
* V ^
* 19 9
* V
* 20 > 21 > 22 > 23 > 24
*
* - Square 3 (because x and y are between -3 and 3 (one of them is always 3 or -3)):
* 36 < 35 < 34 < 33 < 32 < 31 < 30
* V ^
* 37 29
* V ^
* 38 28
* V ^
* 39 27
* V ^
* 40 26
* V ^
* 41 25
* V
* 42 > 43 > 44 > 45 > 46 > 47 > 48
*
* And so on. I call the number of the square "v".
* We can find it by getting the absolute value of both coordinates and get the biggest number.
*
* Then there is the biggest number in the square, the one in the bottom right corner, I call that "g".
* g for square 1 (v = 1) is 8, for v = 2: g = 24, for v = 3: g = 48 and so on.
* We can see that each g is divisble by 8, but the multiplier is slightly tricky to find.
* The multiplier is the sum of all whole number from 1 up until v.
*
* Example:
* v = 2
* g = 8 * (Sum of numbers from 1 to v) = 8 * (1 + 2) = 8 * 3 = 24
*
* v = 3
* g = 8 * (Sum of numbers from 1 to v) = 8 * (1 + 2 + 3) = 8 * 6 = 48
*
* Then, we divide the whole square into two corners. We take the diagonal from top left to bottom right, and check in which part the coordinates are.
* We do that because we have two formulas. In brief, what they do is tranform the coordinates into the number from which we'll subtract from g.
*
* Example:
* v = 3, g = 48
*
* Lets take a look at the formula for the bottom of the square. We aren't interested in g, so the important part is "x - y - 2*v".
* What it does is make the coordinates for x and y correspond to a number from -12 to 0.
* So this:
* 36
* V
* 37
* V
* 38
* V
* 39
* V
* 40
* V
* 41
* V
* 42 > 43 > 44 > 45 > 46 > 47 > 48
* Becomes:
* -12
* V
* -11
* V
* -10
* V
* -9
* V
* -8
* V
* -7
* V
* -6 > -5 > -4 > -3 > -2 > -1 > 0
*
* "- 2 * v" is -6, the number in the bottom left corner. Then, if we increase Y (meaning we want to go up the left column),
* we want the number to also decrease (make it further away from 0). We also have to account for X, because there it's always -3.
* So, we need them to sum up to 0 when we're on the bottom left corner, so we can just subtract Y from X. As we said,
* X is always -3 on the left column, so by increasing Y, we also increase the negative number.
*
* Ok, here are some example numbers:
* X | Y | X - Y
* ---|----|------
* -3 | -3 | 0
* -3 | -2 | -1
* -3 | 0 | -3
*
* This actually works out for us just fine when we're traversing the bottom row. There, Y is always -3, and
* when increasing X we want to increase our number (make it close to 0). So, X - Y become a positive number,
* which summed up with the negative number ("-2*v") makes the result number closer to 0, which is what we want.
*
* Here are more example numbers:
* X | Y | X - Y
* ---|----|------
* -3 | -3 | 0
* -2 | -3 | 1
* 0 | -3 | 3
*
* The formula for the top of the square (top row and right column) works the same , but there our corner number isn't "-2*v" but "-6*v",
* and we want the reverse effect: increase number (make smaller) on increasing X and decerease number on incrasing Y,
* so we make "X - Y" to be "Y - X"
*
* -13 < -14 < -15 < -16 < -17 < -18 < -19
* ^
* -20
* ^
* -21
* ^
* -22
* ^
* -23
* ^
* -24
*/
|
