Bimagic 25x25x25 cube of John-R. Hendricks analyzed

John-R. Hendricks constructed in 2000 the first known bimagic cube (see website http://www.multimagie.com/English/Cube.htm). I have analyzed the bimagic 25x25x25 cube. It is possible to use Hendricks' method and to construct 15625 different bimagic 25x25x25 cubes. One result is even symmetric.

The first grid is based on the following 5x5 square:

		65	65	65	65	65
	65						65
55		1	6	11	16	21
75		15	20	25	5	10		65	65
70		24	4	9	14	19		65	65
65		8	13	18	23	3		65	65
60		17	22	2	7	12		65	65

The 5x5 square vierkant is not fully magic, because not all rows give the magic sum of 65.

The first level of the first grid consists of the 25 shifted versions of the 5x5 squares on a 2x2 carpet. It has the following row-column coordinates (e.g. square 5 - 3 has 2 in the top left corner):

1 - 1   5 - 3 4 - 5   3 - 2   2 - 4
2 - 5   1 - 2   5 - 4   4 - 1   3 - 3
3 - 4   2 - 1   1 - 3   5 - 5   4 - 2
4 - 3   3 - 5   2 - 2   1 - 4   5 - 1
5 - 2   4 - 4   3 - 1   2 - 3   1 - 5

Level 1 up to 25 of the first grid are the shifted versions of the first level with the following row/column-coordinates on a 2x2 carpet of the first level:

1-1, 4-2, 2-3, 5-4, 3-5, 4-5, 2-1, 5-2, 3-3, 1-4, 2-4, 5-5, 3-1, 1-2, 4-3, 5-3, 3-4, 1-5, 4-1, 2-2, 3-2, 1-3, 4-4, 2-5 and 5-1.

The second grid is based on the diagonal shifted version of the 5x5 square which is used for the construction of the first grid:

5x5 square first grid --> diagonal shift second grid

1	6	11	16	21	1	20	9	23	12
15	20	25	5	10	19	8	22	11	5
24	4	9	14	19	7	21	15	4	18
8	13	18	23	3	25	14	3	17	6
17	22	2	7	12	13	2	16	10	24

The 5x5 square of the second grid is a shifted version of a symmetric and panmagic (= ultra magic) 5x5 square! The first level of the second grid consists of the 25 shifted versions of the 5x5 square on a 2x2 carpet. It has the following row-column coordinates (e.g. square 3 - 1 has 7 in the top left corner):

1 - 1   3 - 1   5 - 1   2 - 1   4 - 1
1 - 5   3 - 5   5 - 5   2 - 5   4 - 5
1 - 4   3 - 4   5 - 4   2 - 4   4 - 4
1 - 3   3 - 3   5 - 3   2 - 3   4 - 3
1 - 2   3 - 2   5 - 2   2 - 2   4 - 2

Level 1 up to 25 of the second grid are the shifted versions of the first level with the following row/column-coordinates on a 2x2 carpet of the first level:

1-1, 3-2, 5-3, 2-4, 4-5, 4-3, 1-4, 3-5, 5-1, 2-2, 2-5, 4-1, 1-2, 3-3, 5-4, 5-2, 2-3, 4-4, 1-5, 3-1, 3-4, 5-5, 2-1, 4-2 en 1-3.

The third grid is based on the following 5x5 square:

		15	40	65	90	115
	55						65
65		1	8	15	17	24
65		4	6	13	20	22		65	65
65		2	9	11	18	25		65	75
65		5	7	14	16	23		65	60
65		3	10	12	19	21		65	70

The square vierkant is not fully magic, not all rows and not all (pan)diagonals give the magic sum of 65.

The first level of the third grid consists of the 25 shifted versions of the 5x5 square on a 2x2 carpet. It has the following row-column coordinates (e.g. square 4 - 2 has 7 in the top left corner):

1 - 1   4 - 2   2 - 3   5 - 4   3 - 5
4 - 4   2 - 5   5 - 1   3 - 2   1 - 3
2 - 2   5 - 3   3 - 4   1 - 5   4 - 1
5 - 5   3 - 1   1 - 2   4 - 3   2 - 4
3 - 3   1 - 4   4 - 5   2 - 1   5 - 2

Level 1 up to 25 of the third grid are the shifted versions of the first level with the following row/column-coordinates on a 2x2 carpet of the first level:

1-1, 1-3, 1-5, 1-2, 1-4, 2-4, 2-1, 2-3, 2-5, 2-2, 3-2, 3-4, 3-1, 3-3, 3-5, 4-5, 4-2, 4-4, 4-1, 4-3, 5-3, 5-5, 5-2, 5-4 and 5-1.

It is possible to use in each grid one of the 25 shifted versions of the 5x5 squares. So there are 25x25x25 is 15625 different possibilities to construct a Hendricks' bimagic 25x25x25 cube. It is even possible to construct a symmetric version; see below the 13th (= middle) level:

13th (= middle) level of the symmetric version of Hendricks' bimagic 25x25x25 cube

6875	10955	15060	1040	5145	10021	14851	206	4936	9041	13922	3002	4107	8212	9817	2198	3153	7883	12113	13718	6099	7054	11784	12764	1369
8434	9414	14144	2749	4329	12335	13315	2420	3400	7605	13106	1586	5691	7296	11376	632	5487	6467	11197	15277	4533	9263	10368	14473	428
6893	11748	12703	1808	5913	10794	15524	979	5084	6689	14695	50	4755	8985	10590	2966	3946	8651	9631	13861	3742	7847	11927	13532	2012
9202	10182	14912	267	4497	9978	14083	2563	4168	8273	13129	2359	3339	8069	12174	1405	6135	7240	11345	12950	5301	6281	11011	15241	1221
7661	12391	13496	1951	3556	11562	12542	1647	5852	7457	15463	818	5548	6503	10733	614	4719	8824	10404	14509	3765	8620	9600	14305	2785
129	4984	9089	10069	14799	4030	8135	9865	13970	3075	7926	12031	13636	2241	3221	11827	12807	1287	6017	7122	15103	1083	5188	6793	10898
2463	3443	7548	12253	13358	5739	7344	11449	13029	1509	6390	11245	15350	680	5410	10286	14391	496	4576	9306	14187	2667	4272	8477	9457
922	5002	6732	10837	15567	4823	8903	10508	14738	93	8724	9679	13784	2889	3994	12000	13580	2060	3665	7770	12646	1851	5956	6936	11666
2606	4211	8316	9921	14001	3257	8112	12217	13197	2277	7158	11263	12993	1473	6178	11059	15164	1144	5374	6329	14960	315	4420	9150	10230
1695	5800	7380	11610	12590	5591	6571	10651	15381	861	8867	10472	14552	532	4637	9518	14373	2828	3808	8538	13419	1899	3604	7709	12439
9783	13888	3118	4098	8178	13684	2164	3144	7999	12079	1335	6065	7045	11775	12855	5231	6836	10941	15046	1001	9007	10112	14842	197	4902
11492	13097	1552	5657	7262	15268	748	5453	6433	11163	419	4524	9354	10334	14439	4320	8425	9380	14235	2715	7591	12321	13276	2381	3486
10551	14656	11	4866	8971	13827	2932	3912	8642	9747	2103	3708	7813	11918	13523	5879	6984	11714	12694	1799	6655	10760	15615	970	5075
12140	13245	2350	3305	8035	12911	1391	6246	7201	11306	1187	5292	6272	11102	15207	4463	9193	10173	14878	358	8364	9969	14074	2529	4134
10724	15429	784	5514	6619	14625	580	4685	8790	10395	2771	3851	8581	9561	14291	3547	7627	12482	13462	1942	7448	11528	12508	1738	5843
3187	7917	12022	13727	2207	7088	11818	12798	1253	6108	10989	15094	1074	5154	6759	14765	245	4975	9055	10035	3036	4016	8246	9826	13931
5396	6476	11206	15311	666	9297	10252	14482	462	4567	9448	14153	2633	4363	8468	13349	2429	3409	7514	12369	1625	5705	7310	11415	13020
3960	8690	9670	13775	2980	7856	11961	13566	2046	3626	11632	12737	1842	5947	6902	15533	888	5118	6723	10803	59	4789	8894	10624	14704
6169	7149	11354	12959	1439	6320	11050	15130	1235	5340	10216	14946	276	4381	9236	14117	2597	4177	8282	9887	2268	3373	8078	12183	13163
4728	8833	10438	14543	523	8504	9609	14339	2819	3799	12405	13385	1990	3595	7700	12551	1656	5761	7491	11596	827	5557	6537	10642	15497
12841	1321	6026	7006	11861	1117	5222	6802	10907	15012	4893	9123	10078	14808	163	8169	9774	13979	3084	4064	12070	13675	2130	3235	7965
14405	385	4615	9345	10325	2676	4281	8386	9491	14221	3452	7557	12287	13267	2497	7353	11458	13063	1543	5648	11129	15359	714	5444	6424
13614	2094	3699	7779	11884	1765	5995	6975	11680	12660	5036	6641	10871	15576	931	8937	10542	14647	102	4832	9713	13818	2923	3878	8733
15198	1153	5258	6363	11093	349	4429	9159	10139	14994	4250	8330	9935	14040	2520	8021	12226	13206	2311	3291	11297	12877	1482	6212	7192
14257	2862	3842	8572	9527	1908	3513	7743	12473	13428	5809	7414	11519	12624	1704	6585	10690	15420	775	5605	10481	14586	566	4671	8751

See all grids and all levels of the 25x25x25 bimagic cube in the download below:

25x25x25, Bimagic 25x25x25 cube, grids.x

Microsoft Excel werkblad 1.8 MB

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25x25x25, Bimagic 25x25x25 cube, result.

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25x25x25 magic cube