Harry White shows us on his website http://users.eastlink.ca/~sharrywhite/2xNMultiMagic.html, that it is possible to use
some 8x8 bimagic squares and medjig tiles to construct a bimagic 16x16 square.
Take 1x number from first grid consisting of 4 different Medjig tiles
1 |
3 |
1 |
3 |
4 |
2 |
4 |
2 |
4 |
2 |
4 |
2 |
1 |
3 |
1 |
3 |
2 |
4 |
2 |
4 |
3 |
1 |
3 |
1 |
3 |
1 |
3 |
1 |
2 |
4 |
2 |
4 |
1 |
3 |
3 |
1 |
2 |
4 |
3 |
1 |
4 |
2 |
1 |
3 |
4 |
2 |
2 |
4 |
2 |
4 |
4 |
2 |
1 |
3 |
4 |
2 |
3 |
1 |
2 |
4 |
3 |
1 |
1 |
3 |
1 |
3 |
3 |
1 |
2 |
4 |
3 |
1 |
4 |
2 |
1 |
3 |
4 |
2 |
2 |
4 |
2 |
4 |
4 |
2 |
1 |
3 |
4 |
2 |
3 |
1 |
2 |
4 |
3 |
1 |
1 |
3 |
4 |
2 |
4 |
2 |
1 |
3 |
1 |
3 |
1 |
3 |
1 |
3 |
4 |
2 |
4 |
2 |
3 |
1 |
3 |
1 |
2 |
4 |
2 |
4 |
2 |
4 |
2 |
4 |
3 |
1 |
3 |
1 |
2 |
4 |
2 |
4 |
3 |
1 |
3 |
1 |
3 |
1 |
3 |
1 |
2 |
4 |
2 |
4 |
1 |
3 |
1 |
3 |
4 |
2 |
4 |
2 |
4 |
2 |
4 |
2 |
1 |
3 |
1 |
3 |
4 |
2 |
2 |
4 |
3 |
1 |
2 |
4 |
1 |
3 |
4 |
2 |
1 |
3 |
3 |
1 |
3 |
1 |
1 |
3 |
4 |
2 |
1 |
3 |
2 |
4 |
3 |
1 |
2 |
4 |
4 |
2 |
4 |
2 |
2 |
4 |
3 |
1 |
2 |
4 |
1 |
3 |
4 |
2 |
1 |
3 |
3 |
1 |
3 |
1 |
1 |
3 |
4 |
2 |
1 |
3 |
2 |
4 |
3 |
1 |
2 |
4 |
4 |
2 |
3 |
1 |
3 |
1 |
2 |
4 |
2 |
4 |
2 |
4 |
2 |
4 |
3 |
1 |
3 |
1 |
4 |
2 |
4 |
2 |
1 |
3 |
1 |
3 |
1 |
3 |
1 |
3 |
4 |
2 |
4 |
2 |
+ [number -/- 1] x 4 from second grid (= 2x2 'blown up' 8x8 bimagic square)
3 |
3 |
21 |
21 |
51 |
51 |
14 |
14 |
37 |
37 |
28 |
28 |
62 |
62 |
44 |
44 |
3 |
3 |
21 |
21 |
51 |
51 |
14 |
14 |
37 |
37 |
28 |
28 |
62 |
62 |
44 |
44 |
22 |
22 |
39 |
39 |
1 |
1 |
27 |
27 |
52 |
52 |
42 |
42 |
16 |
16 |
61 |
61 |
22 |
22 |
39 |
39 |
1 |
1 |
27 |
27 |
52 |
52 |
42 |
42 |
16 |
16 |
61 |
61 |
45 |
45 |
32 |
32 |
58 |
58 |
36 |
36 |
11 |
11 |
17 |
17 |
55 |
55 |
6 |
6 |
45 |
45 |
32 |
32 |
58 |
58 |
36 |
36 |
11 |
11 |
17 |
17 |
55 |
55 |
6 |
6 |
40 |
40 |
50 |
50 |
24 |
24 |
41 |
41 |
2 |
2 |
63 |
63 |
25 |
25 |
15 |
15 |
40 |
40 |
50 |
50 |
24 |
24 |
41 |
41 |
2 |
2 |
63 |
63 |
25 |
25 |
15 |
15 |
60 |
60 |
46 |
46 |
12 |
12 |
53 |
53 |
30 |
30 |
35 |
35 |
5 |
5 |
19 |
19 |
60 |
60 |
46 |
46 |
12 |
12 |
53 |
53 |
30 |
30 |
35 |
35 |
5 |
5 |
19 |
19 |
49 |
49 |
4 |
4 |
38 |
38 |
64 |
64 |
23 |
23 |
13 |
13 |
43 |
43 |
26 |
26 |
49 |
49 |
4 |
4 |
38 |
38 |
64 |
64 |
23 |
23 |
13 |
13 |
43 |
43 |
26 |
26 |
10 |
10 |
59 |
59 |
29 |
29 |
7 |
7 |
48 |
48 |
54 |
54 |
20 |
20 |
33 |
33 |
10 |
10 |
59 |
59 |
29 |
29 |
7 |
7 |
48 |
48 |
54 |
54 |
20 |
20 |
33 |
33 |
31 |
31 |
9 |
9 |
47 |
47 |
18 |
18 |
57 |
57 |
8 |
8 |
34 |
34 |
56 |
56 |
31 |
31 |
9 |
9 |
47 |
47 |
18 |
18 |
57 |
57 |
8 |
8 |
34 |
34 |
56 |
56 |
Bimagic 16x16 square
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
2056 |
|||
2056 |
2056 |
|||||||||||||||||
2056 |
9 |
11 |
81 |
83 |
204 |
202 |
56 |
54 |
148 |
146 |
112 |
110 |
245 |
247 |
173 |
175 |
||
2056 |
10 |
12 |
82 |
84 |
203 |
201 |
55 |
53 |
147 |
145 |
111 |
109 |
246 |
248 |
174 |
176 |
||
2056 |
85 |
87 |
155 |
153 |
2 |
4 |
107 |
105 |
208 |
206 |
165 |
167 |
64 |
62 |
242 |
244 |
||
2056 |
86 |
88 |
156 |
154 |
1 |
3 |
108 |
106 |
207 |
205 |
166 |
168 |
63 |
61 |
241 |
243 |
||
2056 |
177 |
179 |
127 |
125 |
230 |
232 |
143 |
141 |
44 |
42 |
65 |
67 |
220 |
218 |
22 |
24 |
||
2056 |
178 |
180 |
128 |
126 |
229 |
231 |
144 |
142 |
43 |
41 |
66 |
68 |
219 |
217 |
21 |
23 |
||
2056 |
160 |
158 |
200 |
198 |
93 |
95 |
161 |
163 |
5 |
7 |
249 |
251 |
100 |
98 |
60 |
58 |
||
2056 |
159 |
157 |
199 |
197 |
94 |
96 |
162 |
164 |
6 |
8 |
250 |
252 |
99 |
97 |
59 |
57 |
||
2056 |
238 |
240 |
182 |
184 |
47 |
45 |
211 |
209 |
119 |
117 |
139 |
137 |
18 |
20 |
74 |
76 |
||
2056 |
237 |
239 |
181 |
183 |
48 |
46 |
212 |
210 |
120 |
118 |
140 |
138 |
17 |
19 |
73 |
75 |
||
2056 |
196 |
194 |
14 |
16 |
151 |
149 |
254 |
256 |
89 |
91 |
52 |
50 |
169 |
171 |
103 |
101 |
||
2056 |
195 |
193 |
13 |
15 |
152 |
150 |
253 |
255 |
90 |
92 |
51 |
49 |
170 |
172 |
104 |
102 |
||
2056 |
40 |
38 |
234 |
236 |
115 |
113 |
26 |
28 |
189 |
191 |
216 |
214 |
77 |
79 |
131 |
129 |
||
2056 |
39 |
37 |
233 |
235 |
116 |
114 |
25 |
27 |
190 |
192 |
215 |
213 |
78 |
80 |
132 |
130 |
||
2056 |
123 |
121 |
35 |
33 |
186 |
188 |
70 |
72 |
226 |
228 |
30 |
32 |
135 |
133 |
223 |
221 |
||
2056 |
124 |
122 |
36 |
34 |
185 |
187 |
69 |
71 |
225 |
227 |
29 |
31 |
136 |
134 |
224 |
222 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
351576 |
|||
351576 |
351576 |
|||||||||||||||||
351576 |
81 |
121 |
6561 |
6889 |
41616 |
40804 |
3136 |
2916 |
21904 |
21316 |
12544 |
12100 |
60025 |
61009 |
29929 |
30625 |
||
351576 |
100 |
144 |
6724 |
7056 |
41209 |
40401 |
3025 |
2809 |
21609 |
21025 |
12321 |
11881 |
60516 |
61504 |
30276 |
30976 |
||
351576 |
7225 |
7569 |
24025 |
23409 |
4 |
16 |
11449 |
11025 |
43264 |
42436 |
27225 |
27889 |
4096 |
3844 |
58564 |
59536 |
||
351576 |
7396 |
7744 |
24336 |
23716 |
1 |
9 |
11664 |
11236 |
42849 |
42025 |
27556 |
28224 |
3969 |
3721 |
58081 |
59049 |
||
351576 |
31329 |
32041 |
16129 |
15625 |
52900 |
53824 |
20449 |
19881 |
1936 |
1764 |
4225 |
4489 |
48400 |
47524 |
484 |
576 |
||
351576 |
31684 |
32400 |
16384 |
15876 |
52441 |
53361 |
20736 |
20164 |
1849 |
1681 |
4356 |
4624 |
47961 |
47089 |
441 |
529 |
||
351576 |
25600 |
24964 |
40000 |
39204 |
8649 |
9025 |
25921 |
26569 |
25 |
49 |
62001 |
63001 |
10000 |
9604 |
3600 |
3364 |
||
351576 |
25281 |
24649 |
39601 |
38809 |
8836 |
9216 |
26244 |
26896 |
36 |
64 |
62500 |
63504 |
9801 |
9409 |
3481 |
3249 |
||
351576 |
56644 |
57600 |
33124 |
33856 |
2209 |
2025 |
44521 |
43681 |
14161 |
13689 |
19321 |
18769 |
324 |
400 |
5476 |
5776 |
||
351576 |
56169 |
57121 |
32761 |
33489 |
2304 |
2116 |
44944 |
44100 |
14400 |
13924 |
19600 |
19044 |
289 |
361 |
5329 |
5625 |
||
351576 |
38416 |
37636 |
196 |
256 |
22801 |
22201 |
64516 |
65536 |
7921 |
8281 |
2704 |
2500 |
28561 |
29241 |
10609 |
10201 |
||
351576 |