Special magic squares

Besides the ordinary magic squares, where the sum of the numbers in the rows, columns and diagonals is equal, there are also magic squares where this is true for the product of the numbers. You can easily find them yourself. I will show you how later on.
There are also squares, where within each row, column or diagonal, the sum of the square (or higher power) of the numbers is always equal. These magic squares are more difficult to find than the ordinary ones and sometimes they exist only from a certain order of the square.
☞ The order of a square is the number of rows (and thus also the number of columns).

Squares with squared numbers

In these squares, the numbers themselves are not magic, but the squares of the numbers are.
Here is an example:

68²	29²	41²	37²
17²	31²	79²	32²
59²	28²	23²	61²
11²	77²	8²	49²

4624	841	1681	1369
289	961	6241	1024
3481	784	529	3721
121	5929	64	2401

Sum of the squares in each row, column or diagonal: 8515

Bimagic squares

If the numbers themselves also form a magic square, this is called a bimagic square. It is assumed that there are no bimagic squares with an order smaller than 8.
Here is an example:

16	41	36	5	27	62	55	18
26	63	54	19	13	44	33	8
1	40	45	12	22	51	58	31
23	50	59	30	4	37	48	9
38	3	10	47	49	24	29	60
52	21	32	57	39	2	11	46
43	14	7	34	64	25	20	53
61	28	17	56	42	15	6	35

Sum of the squares in each row, column or diagonal: 260

256	1681	1296	25	729	3844	3025	324
676	3969	2916	361	169	1936	1089	64
1	1600	2025	144	484	2601	3364	961
529	2500	3481	900	16	1369	2304	81
1444	9	100	2209	2401	576	841	3600
2704	441	1024	3249	1521	4	121	2116
1849	196	49	1156	4096	625	400	2809
3721	784	289	3136	1764	225	36	1225

Sum of the squares in each row, column or diagonal: 11180

Trimagic squares

There are trimagic squares. These are squares that remain magic if all the numbers are squared or taken to the third power. Only trimagic squares of order 12, 32, 64, 81 and 128 are known. I have one of order 12.

1	9	75	74	140	122	55	132	73	58	80	51
22	119	141	8	101	76	27	117	64	98	34	63
33	45	35	106	124	142	95	68	2	84	105	31
41	115	48	49	42	86	135	91	121	116	6	20
62	107	57	12	60	67	130	11	109	138	92	25
66	93	14	43	37	126	89	99	32	16	127	128
79	52	131	102	108	19	56	46	113	129	18	17
83	38	88	133	85	78	15	134	36	7	53	120
104	30	97	96	103	59	10	54	24	29	139	125
112	100	110	39	21	3	50	77	143	61	40	114
123	26	4	137	44	69	118	28	81	47	111	82
144	136	70	71	5	23	90	13	72	87	65	94

The sum of the squares in each row, column or diagonal is 870

1²	9²	75²	74²	140²	122²	55²	132²	73²	58²	80²	51²
22²	119²	141²	8²	101²	76²	27²	117²	64²	98²	34²	63²
33²	45²	35²	106²	124²	142²	95²	68²	2²	84²	105²	31²
41²	115²	48²	49²	42²	86²	135²	91²	121²	116²	6²	20²
62²	107²	57²	12²	60²	67²	130²	11²	109²	138²	92²	25²
66²	93²	14²	43²	37²	126²	89²	99²	32²	16²	127²	128²
79²	52²	131²	102²	108²	19²	56²	46²	113²	129²	18²	17²
83²	38²	88²	133²	85²	78²	15²	134²	36²	7²	53²	120²
104²	30²	97²	96²	103²	59²	10²	54²	24²	29²	139²	125²
112²	100²	110²	39²	21²	3²	50²	77²	143²	61²	40²	114²
123²	26²	4²	137²	44²	69²	118²	28²	81²	47²	111²	82²
144²	136²	70²	71²	5²	23²	90²	13²	72²	87²	65²	94²

1	81	5625	5476	19600	14884	3025	17424	5329	3364	6400	2601
484	14161	19881	64	10201	5776	729	13689	4096	9604	1156	3969
1089	2025	1225	11236	15376	20164	9025	4624	4	7056	11025	961
1681	13225	2304	2401	1764	7396	18225	8281	14641	13456	36	400
3844	11449	3249	144	3600	4489	16900	121	11881	19044	8464	625
4356	8649	196	1849	1369	15876	7921	9801	1024	256	16129	16384
6241	2704	17161	10404	11664	361	3136	2116	12769	16641	324	289
6889	1444	7744	17689	7225	6084	225	17956	1296	49	2809	14400
10816	900	9409	9216	10609	3481	100	2916	576	841	19321	15625
12544	10000	12100	1521	441	9	2500	5929	20449	3721	1600	12996
15129	676	16	18769	1936	4761	13924	784	6561	2209	12321	6724
20736	18496	4900	5041	25	529	8100	169	5184	7569	4225	8836

The sum of the squares in each row, column or diagonal is 83810

1³	9³	75³	74³	140³	122³	55³	132³	73³	58³	80³	51³
22³	119³	141³	8³	101³	76³	27³	117³	64³	98³	34³	63³
33³	45³	35³	106³	124³	142³	95³	68³	2³	84³	105³	31³
41³	115³	48³	49³	42³	86³	135³	91³	121³	116³	6³	20³
62³	107³	57³	12³	60³	67³	130³	11³	109³	138³	92³	25³
66³	93³	14³	43³	37³	126³	89³	99³	32³	16³	127³	128³
79³	52³	131³	102³	108³	19³	56³	46³	113³	129³	18³	17³
83³	38³	88³	133³	85³	78³	15³	134³	36³	7³	53³	120³
104³	30³	97³	96³	103³	59³	10³	54³	24³	29³	139³	125³
112³	100³	110³	39³	21³	3³	50³	77³	143³	61³	40³	114³
123³	26³	4³	137³	44³	69³	118³	28³	81³	47³	111³	82³
144³	136³	70³	71³	5³	23³	90³	13³	72³	87³	65³	94³

1	729	421875	405224	2744000	1815848	166375	2299968	389017	195112	512000	132651
10648	1685159	2803221	512	1030301	438976	19683	1601613	262144	941192	39304	250047
35937	91125	42875	1191016	1906624	2863288	857375	314432	8	592704	1157625	29791
68921	1520875	110592	117649	74088	636056	2460375	753571	1771561	1560896	216	8000
238328	1225043	185193	1728	216000	300763	2197000	1331	1295029	2628072	778688	15625
287496	804357	2744	79507	50653	2000376	704969	970299	32768	4096	2048383	2097152
493039	140608	2248091	1061208	1259712	6859	175616	97336	1442897	2146689	5832	4913
571787	54872	681472	2352637	614125	474552	3375	2406104	46656	343	148877	1728000
1124864	27000	912673	884736	1092727	205379	1000	157464	13824	24389	2685619	1953125
1404928	1000000	1331000	59319	9261	27	125000	456533	2924207	226981	64000	1481544
1860867	17576	64	2571353	85184	328509	1643032	21952	531441	103823	1367631	551368
2985984	2515456	343000	357911	125	12167	729000	2197	373248	658503	274625	830584

The sum in each row, column or diagonal is 9082800

Magic squares with × instead of +

If, instead of adding the numbers, you multiply them, you can also get a magic square. The product of all the numbers in the rows, columns and diagonals is then the same. Here is an example:

256

128

You can make a multiplication magic square yourself very easily. Make an ordinary magic square, possibly with my programme on this page. Choose a base number, for example 2. Replace all the numbers in the magic square with that base to the power (number in the diagram). The square above is derived from the following ordinary magic square by choosing ground number 2.

The magic constant in this case is 12.

We can make another multiplication magic square by choosing base number 3:

3¹

3⁸

3³

3⁶

3⁴

3²

3⁵

3⁰

3⁷

→

6561

729

243

2187

So from an ordinary magic square, you can make infinitely many different multiplication magic squares. The magic multiplication constant is easy to calculate: the base number to the power (magic constant of the ordinary square). So for the square above: 3¹² = 531441.

Magic square for both sum and product

That is also possible: the sum in each row, column and diagonal is 840, the product is 2058068231856000.

46	81	117	102	15	76	200	203
19	60	232	175	54	69	153	78
216	161	17	52	171	90	58	75
135	114	50	87	184	189	13	68
150	261	45	38	91	136	92	27
119	104	108	23	174	225	57	30
116	25	133	120	51	26	162	207
39	34	138	243	100	29	105	152

16	41	36	5	27	62	55	18
26	63	54	19	13	44	33	8
1	40	45	12	22	51	58	31
23	50	59	30	4	37	48	9
38	3	10	47	49	24	29	60
52	21	32	57	39	2	11	46
43	14	7	34	64	25	20	53
61	28	17	56	42	15	6	35

1	9	75	74	140	122	55	132	73	58	80	51
22	119	141	8	101	76	27	117	64	98	34	63
33	45	35	106	124	142	95	68	2	84	105	31
41	115	48	49	42	86	135	91	121	116	6	20
62	107	57	12	60	67	130	11	109	138	92	25
66	93	14	43	37	126	89	99	32	16	127	128
79	52	131	102	108	19	56	46	113	129	18	17
83	38	88	133	85	78	15	134	36	7	53	120
104	30	97	96	103	59	10	54	24	29	139	125
112	100	110	39	21	3	50	77	143	61	40	114
123	26	4	137	44	69	118	28	81	47	111	82
144	136	70	71	5	23	90	13	72	87	65	94

46	81	117	102	15	76	200	203
19	60	232	175	54	69	153	78
216	161	17	52	171	90	58	75
135	114	50	87	184	189	13	68
150	261	45	38	91	136	92	27
119	104	108	23	174	225	57	30
116	25	133	120	51	26	162	207
39	34	138	243	100	29	105	152

16	41	36	5	27	62	55	18
26	63	54	19	13	44	33	8
1	40	45	12	22	51	58	31
23	50	59	30	4	37	48	9
38	3	10	47	49	24	29	60
52	21	32	57	39	2	11	46
43	14	7	34	64	25	20	53
61	28	17	56	42	15	6	35

1	9	75	74	140	122	55	132	73	58	80	51
22	119	141	8	101	76	27	117	64	98	34	63
33	45	35	106	124	142	95	68	2	84	105	31
41	115	48	49	42	86	135	91	121	116	6	20
62	107	57	12	60	67	130	11	109	138	92	25
66	93	14	43	37	126	89	99	32	16	127	128
79	52	131	102	108	19	56	46	113	129	18	17
83	38	88	133	85	78	15	134	36	7	53	120
104	30	97	96	103	59	10	54	24	29	139	125
112	100	110	39	21	3	50	77	143	61	40	114
123	26	4	137	44	69	118	28	81	47	111	82
144	136	70	71	5	23	90	13	72	87	65	94

46	81	117	102	15	76	200	203
19	60	232	175	54	69	153	78
216	161	17	52	171	90	58	75
135	114	50	87	184	189	13	68
150	261	45	38	91	136	92	27
119	104	108	23	174	225	57	30
116	25	133	120	51	26	162	207
39	34	138	243	100	29	105	152

16	41	36	5	27	62	55	18
26	63	54	19	13	44	33	8
1	40	45	12	22	51	58	31
23	50	59	30	4	37	48	9
38	3	10	47	49	24	29	60
52	21	32	57	39	2	11	46
43	14	7	34	64	25	20	53
61	28	17	56	42	15	6	35

1	9	75	74	140	122	55	132	73	58	80	51
22	119	141	8	101	76	27	117	64	98	34	63
33	45	35	106	124	142	95	68	2	84	105	31
41	115	48	49	42	86	135	91	121	116	6	20
62	107	57	12	60	67	130	11	109	138	92	25
66	93	14	43	37	126	89	99	32	16	127	128
79	52	131	102	108	19	56	46	113	129	18	17
83	38	88	133	85	78	15	134	36	7	53	120
104	30	97	96	103	59	10	54	24	29	139	125
112	100	110	39	21	3	50	77	143	61	40	114
123	26	4	137	44	69	118	28	81	47	111	82
144	136	70	71	5	23	90	13	72	87	65	94

46	81	117	102	15	76	200	203
19	60	232	175	54	69	153	78
216	161	17	52	171	90	58	75
135	114	50	87	184	189	13	68
150	261	45	38	91	136	92	27
119	104	108	23	174	225	57	30
116	25	133	120	51	26	162	207
39	34	138	243	100	29	105	152