6)
Every
other row, column and diagonal of x2 + xy + y2 has a
minimum value, which all work together to form lines passing through 0.
Every other
row, column and diagonal of x2 + xy + y2 has a minimum
value, which all work together to form lines passing through 0.
-10
|
-9
|
-8
|
-7
|
-6
|
-5
|
-4
|
-3
|
-2
|
-1
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
||
10
|
100
|
91
|
84
|
79
|
76
|
75
|
76
|
79
|
84
|
91
|
100
|
111
|
124
|
139
|
156
|
175
|
196
|
219
|
244
|
271
|
300
|
10
|
9
|
91
|
81
|
73
|
67
|
63
|
61
|
61
|
63
|
67
|
73
|
81
|
91
|
103
|
117
|
133
|
151
|
171
|
193
|
217
|
243
|
271
|
9
|
8
|
84
|
73
|
64
|
57
|
52
|
49
|
48
|
49
|
52
|
57
|
64
|
73
|
84
|
97
|
112
|
129
|
148
|
169
|
192
|
217
|
244
|
8
|
7
|
79
|
67
|
57
|
49
|
43
|
39
|
37
|
37
|
39
|
43
|
49
|
57
|
67
|
79
|
93
|
109
|
127
|
147
|
169
|
193
|
219
|
7
|
6
|
76
|
63
|
52
|
43
|
36
|
31
|
28
|
27
|
28
|
31
|
36
|
43
|
52
|
63
|
76
|
91
|
108
|
127
|
148
|
171
|
196
|
6
|
5
|
75
|
61
|
49
|
39
|
31
|
25
|
21
|
19
|
19
|
21
|
25
|
31
|
39
|
49
|
61
|
75
|
91
|
109
|
129
|
151
|
175
|
5
|
4
|
76
|
61
|
48
|
37
|
28
|
21
|
16
|
13
|
12
|
13
|
16
|
21
|
28
|
37
|
48
|
61
|
76
|
93
|
112
|
133
|
156
|
4
|
3
|
79
|
63
|
49
|
37
|
27
|
19
|
13
|
9
|
7
|
7
|
9
|
13
|
19
|
27
|
37
|
49
|
63
|
79
|
97
|
117
|
139
|
3
|
2
|
84
|
67
|
52
|
39
|
28
|
19
|
12
|
7
|
4
|
3
|
4
|
7
|
12
|
19
|
28
|
39
|
52
|
67
|
84
|
103
|
124
|
2
|
1
|
91
|
73
|
57
|
43
|
31
|
21
|
13
|
7
|
3
|
1
|
1
|
3
|
7
|
13
|
21
|
31
|
43
|
57
|
73
|
91
|
111
|
1
|
0
|
100
|
81
|
64
|
49
|
36
|
25
|
16
|
9
|
4
|
1
|
0
|
1
|
4
|
9
|
16
|
25
|
36
|
49
|
64
|
81
|
100
|
0
|
-1
|
111
|
91
|
73
|
57
|
43
|
31
|
21
|
13
|
7
|
3
|
1
|
1
|
3
|
7
|
13
|
21
|
31
|
43
|
57
|
73
|
91
|
-1
|
-2
|
124
|
103
|
84
|
67
|
52
|
39
|
28
|
19
|
12
|
7
|
4
|
3
|
4
|
7
|
12
|
19
|
28
|
39
|
52
|
67
|
84
|
-2
|
-3
|
139
|
117
|
97
|
79
|
63
|
49
|
37
|
27
|
19
|
13
|
9
|
7
|
7
|
9
|
13
|
19
|
27
|
37
|
49
|
63
|
79
|
-3
|
-4
|
156
|
133
|
112
|
93
|
76
|
61
|
48
|
37
|
28
|
21
|
16
|
13
|
12
|
13
|
16
|
21
|
28
|
37
|
48
|
61
|
76
|
-4
|
-5
|
175
|
151
|
129
|
109
|
91
|
75
|
61
|
49
|
39
|
31
|
25
|
21
|
19
|
19
|
21
|
25
|
31
|
39
|
49
|
61
|
75
|
-5
|
-6
|
196
|
171
|
148
|
127
|
108
|
91
|
76
|
63
|
52
|
43
|
36
|
31
|
28
|
27
|
28
|
31
|
36
|
43
|
52
|
63
|
76
|
-6
|
-7
|
219
|
193
|
169
|
147
|
127
|
109
|
93
|
79
|
67
|
57
|
49
|
43
|
39
|
37
|
37
|
39
|
43
|
49
|
57
|
67
|
79
|
-7
|
-8
|
244
|
217
|
192
|
169
|
148
|
129
|
112
|
97
|
84
|
73
|
64
|
57
|
52
|
49
|
48
|
49
|
52
|
57
|
64
|
73
|
84
|
-8
|
-9
|
271
|
243
|
217
|
193
|
171
|
151
|
133
|
117
|
103
|
91
|
81
|
73
|
67
|
63
|
61
|
61
|
63
|
67
|
73
|
81
|
91
|
-9
|
-10
|
300
|
271
|
244
|
219
|
196
|
175
|
156
|
139
|
124
|
111
|
100
|
91
|
84
|
79
|
76
|
75
|
76
|
79
|
84
|
91
|
100
|
-10
|
-10
|
-9
|
-8
|
-7
|
-6
|
-5
|
-4
|
-3
|
-2
|
-1
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
On even
rows, where y = 2n, so x2 + xy + y2 = x2 + 2nx
+ 4n2. The minimum would be
the vertex x = -b/(2a) = -2n/(2∙1) = -n, so each minimum goes through (–n, 2n)
which can be represented by y = -2x, a line going through (0, 0). A similar argument can be made for even
columns, but switching x and y so that each minimum goes through (2n, -n) which
can be represented by y = -½x, which is also a line going through (0, 0). On even diagonals of negative slope, x + y =
2n, so y = -x + 2n, so x2 + xy + y2 = x2 +
x(-x + 2n) + (-x + 2n)2 = x2 – x2 + 2nx + x2
– 4nx + 4n2 = x2 – 2nx + 4n2. The minimum would be the vertex x = -b/(2a) =
-(-2n)/(2∙1) = n, so y = -x + 2n = -n + 2n = n, so each minimum goes through
(n, n) which can be represented by y = x, yet another line going through (0,
0). Finally, on even diagonals of
positive slope, y – x = 2n, so y = x + 2n, so x2 + xy + y2
= x2 + x(x + 2n) + (x + 2n)2 = x2 + x2
+ 2nx + x2 + 4nx + 4n2 = 3x2 + 6nx + 4n2. The minimum would be the vertex x = -b/(2a) =
-(6n)/(2∙3) = -n, so y = x + 2n = -n + 2n = n, so each minimum goes through
(-n, n) which can be represented by y = -x, yet another line going through (0,
0).
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