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Other 4N+0 Magic Square Sizes:
Detailed Directions for: 4x4 * 8x8 * 12x12 *
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* 16 * 20 * 24 * 28 * 32 * 36 * 40 * 44 * 48 * 52 * 56 *
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A 68x68 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (157250)

A Magic Square is a square of numbers in which every row, every column, and both diagonals add up to the same number. This number is often called the magic sum. The 68 by 68 magic square shown below has a magic sum equal to 157250.
Though magic squares can be made with non-consecutive and non-regular sequences, they are usually seen made up of consecutive numbers, possibly because it is harder to be consecutive.

The method described below can be used to create all 4N+0 (multiple-of-four) magic squares. This is just one method for creating magic squares, there are many other methods, and many-many-many magic squares to be found.

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How to Create a 68x68 Magic Square
with a Magic Sum of 157250

The method described on this page can be used to create 4x4, 8x8, 12x12, 16x16 and larger 4N+0 magic squares

Steps:

Draw a 68x68 blank grid and color in the two main diagonals.
Also, color in the two main diagonals of each 4x4 box within this 68x68 square.
Starting in the top left corner, moving left to right through the rows, count the squares, 1, 2, 3, 4, but only write down the numbers you say when you get to one of the colored in squares. If you do this properly, you will write 1 in the top left corner and write 4624 in the bottom right corner.
Next, do the same thing, starting in the bottom right corner, moving right-to-left upward through the rows, again count from 1 to 4624, but this time, only write down the numbers in the non-shaded squares.

Step One: Draw a 68x68 blank grid and
shade in the two main diagonals

You don't necessarily have to shade in the squares. You could simply draw a light line that passes through these squares. You only need to be able to tell the difference between squares that are on these 4x4 diagonals and those that are not.

Step Two: Shade in all the 4x4 diagonals

If you are making a square larger than 4x4, the entire square can be visualized as a collection of 4x4 squares. Shade in the main diagonals of each 4x4 square within the big square.

If you do this correctly, by the time you are done, you will have shaded in exactly half the squares in your overall grid. If you had simply drawn diagonal lines through those squares, you would see a series of X's.

Step Three: Count the squares
(writing only on the shaded squares)

This is fairly straightforward. Starting in the top left corner, and moving left-to-right through each row, count the squares, but only write down the numbers in the shaded squares, but remember to count all the blank squares. In the first four spaces, you would write down 1, count 2 and 3, and write down 4. Continue until you have written down the last number (4624).

Step Four: Count the squares again, counting backward,
writing into the blank squares

This time, you can either

Start counting backward, starting with 4624 (which you would not write down) and count downward, writing down the number whenever you come to a blank square. The last two numbers you write down would be 3 and 2 just before you get to the 4624 which you wrote down in Step Three.
Or, if you prefer counting upward, start with 1 in the bottom right corner and move right-to-left through each row until you get to the top left corner. Working this way, you will skip 1 (because 4624 is already written in that square), write down 2 and 3, then skip 4, etc.

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A 68-by-68 Magic Square

(Magic Sum = 157250)

1	4623	4622	4	5	4619	4618	8	9	4615	4614	12	13	4611	4610	16	17	4607	4606	20	21	4603	4602	24	25	4599	4598	28	29	4595	4594	32	33	4591	4590	36	37	4587	4586	40	41	4583	4582	44	45	4579	4578	48	49	4575	4574	52	53	4571	4570	56	57	4567	4566	60	61	4563	4562	64	65	4559	4558	68
4556	70	71	4553	4552	74	75	4549	4548	78	79	4545	4544	82	83	4541	4540	86	87	4537	4536	90	91	4533	4532	94	95	4529	4528	98	99	4525	4524	102	103	4521	4520	106	107	4517	4516	110	111	4513	4512	114	115	4509	4508	118	119	4505	4504	122	123	4501	4500	126	127	4497	4496	130	131	4493	4492	134	135	4489
4488	138	139	4485	4484	142	143	4481	4480	146	147	4477	4476	150	151	4473	4472	154	155	4469	4468	158	159	4465	4464	162	163	4461	4460	166	167	4457	4456	170	171	4453	4452	174	175	4449	4448	178	179	4445	4444	182	183	4441	4440	186	187	4437	4436	190	191	4433	4432	194	195	4429	4428	198	199	4425	4424	202	203	4421
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3400	1226	1227	3397	3396	1230	1231	3393	3392	1234	1235	3389	3388	1238	1239	3385	3384	1242	1243	3381	3380	1246	1247	3377	3376	1250	1251	3373	3372	1254	1255	3369	3368	1258	1259	3365	3364	1262	1263	3361	3360	1266	1267	3357	3356	1270	1271	3353	3352	1274	1275	3349	3348	1278	1279	3345	3344	1282	1283	3341	3340	1286	1287	3337	3336	1290	1291	3333
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3128	1498	1499	3125	3124	1502	1503	3121	3120	1506	1507	3117	3116	1510	1511	3113	3112	1514	1515	3109	3108	1518	1519	3105	3104	1522	1523	3101	3100	1526	1527	3097	3096	1530	1531	3093	3092	1534	1535	3089	3088	1538	1539	3085	3084	1542	1543	3081	3080	1546	1547	3077	3076	1550	1551	3073	3072	1554	1555	3069	3068	1558	1559	3065	3064	1562	1563	3061
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1836	2790	2791	1833	1832	2794	2795	1829	1828	2798	2799	1825	1824	2802	2803	1821	1820	2806	2807	1817	1816	2810	2811	1813	1812	2814	2815	1809	1808	2818	2819	1805	1804	2822	2823	1801	1800	2826	2827	1797	1796	2830	2831	1793	1792	2834	2835	1789	1788	2838	2839	1785	1784	2842	2843	1781	1780	2846	2847	1777	1776	2850	2851	1773	1772	2854	2855	1769
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2925	1699	1698	2928	2929	1695	1694	2932	2933	1691	1690	2936	2937	1687	1686	2940	2941	1683	1682	2944	2945	1679	1678	2948	2949	1675	1674	2952	2953	1671	1670	2956	2957	1667	1666	2960	2961	1663	1662	2964	2965	1659	1658	2968	2969	1655	1654	2972	2973	1651	1650	2976	2977	1647	1646	2980	2981	1643	1642	2984	2985	1639	1638	2988	2989	1635	1634	2992
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4081	543	542	4084	4085	539	538	4088	4089	535	534	4092	4093	531	530	4096	4097	527	526	4100	4101	523	522	4104	4105	519	518	4108	4109	515	514	4112	4113	511	510	4116	4117	507	506	4120	4121	503	502	4124	4125	499	498	4128	4129	495	494	4132	4133	491	490	4136	4137	487	486	4140	4141	483	482	4144	4145	479	478	4148
476	4150	4151	473	472	4154	4155	469	468	4158	4159	465	464	4162	4163	461	460	4166	4167	457	456	4170	4171	453	452	4174	4175	449	448	4178	4179	445	444	4182	4183	441	440	4186	4187	437	436	4190	4191	433	432	4194	4195	429	428	4198	4199	425	424	4202	4203	421	420	4206	4207	417	416	4210	4211	413	412	4214	4215	409
408	4218	4219	405	404	4222	4223	401	400	4226	4227	397	396	4230	4231	393	392	4234	4235	389	388	4238	4239	385	384	4242	4243	381	380	4246	4247	377	376	4250	4251	373	372	4254	4255	369	368	4258	4259	365	364	4262	4263	361	360	4266	4267	357	356	4270	4271	353	352	4274	4275	349	348	4278	4279	345	344	4282	4283	341
4285	339	338	4288	4289	335	334	4292	4293	331	330	4296	4297	327	326	4300	4301	323	322	4304	4305	319	318	4308	4309	315	314	4312	4313	311	310	4316	4317	307	306	4320	4321	303	302	4324	4325	299	298	4328	4329	295	294	4332	4333	291	290	4336	4337	287	286	4340	4341	283	282	4344	4345	279	278	4348	4349	275	274	4352
4353	271	270	4356	4357	267	266	4360	4361	263	262	4364	4365	259	258	4368	4369	255	254	4372	4373	251	250	4376	4377	247	246	4380	4381	243	242	4384	4385	239	238	4388	4389	235	234	4392	4393	231	230	4396	4397	227	226	4400	4401	223	222	4404	4405	219	218	4408	4409	215	214	4412	4413	211	210	4416	4417	207	206	4420
204	4422	4423	201	200	4426	4427	197	196	4430	4431	193	192	4434	4435	189	188	4438	4439	185	184	4442	4443	181	180	4446	4447	177	176	4450	4451	173	172	4454	4455	169	168	4458	4459	165	164	4462	4463	161	160	4466	4467	157	156	4470	4471	153	152	4474	4475	149	148	4478	4479	145	144	4482	4483	141	140	4486	4487	137
136	4490	4491	133	132	4494	4495	129	128	4498	4499	125	124	4502	4503	121	120	4506	4507	117	116	4510	4511	113	112	4514	4515	109	108	4518	4519	105	104	4522	4523	101	100	4526	4527	97	96	4530	4531	93	92	4534	4535	89	88	4538	4539	85	84	4542	4543	81	80	4546	4547	77	76	4550	4551	73	72	4554	4555	69
4557	67	66	4560	4561	63	62	4564	4565	59	58	4568	4569	55	54	4572	4573	51	50	4576	4577	47	46	4580	4581	43	42	4584	4585	39	38	4588	4589	35	34	4592	4593	31	30	4596	4597	27	26	4600	4601	23	22	4604	4605	19	18	4608	4609	15	14	4612	4613	11	10	4616	4617	7	6	4620	4621	3	2	4624

This Magic Square was created by John Knoderer, a webmaster and computer programmer who is available to telecommute to your location. (In other words, John would be delighted to create puzzles for your website or do other amazing work for you on a contract basis.)

If you found this page useful, and especially if you printed this page to distribute to students, we invite you to send a suitable donation to John (address below) or PayPal your donation to Donations (at) Mazes.com.

You can find more of John's puzzling materials at www.MAZES.com and at www.GodLovesEveryone.org. You can reach John at Webmaster (at) Mazes.com.

Fine Print: If you print and duplicate these magic squares for classroom use or other multiple copy use, please send a payment of $(however much you think this is worth) to the author. John puts a GREAT DEAL of time into creating and posting materials to the internet for you to use (and enjoy). Your donations will help pay bills and keep new materials coming your way.

Send your donations to:

John Knoderer
www.MAZES.com
P O Box 235
Sulphur Springs, AR 72768 USA

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E-mail the webmaster: webmaster (at) mazes.com

The contents of this page are copyright 2005 by the author. However, we recognize that teachers may wish to download and use our materials with their students. We give that permission, as long as the teacher sends what they believe to be a fair donation to the author each time the material is used.

A 68x68 Magic Square

Scroll down to see a magic square in which all rows, columns, and both diagonals add up to the same magic sum (157250)

Read more about Magic Squares

How to Create a 68x68 Magic Square with a Magic Sum of 157250

Step One: Draw a 68x68 blank grid and shade in the two main diagonals

Step Two: Shade in all the 4x4 diagonals

Step Three: Count the squares (writing only on the shaded squares)

Step Four: Count the squares again, counting backward, writing into the blank squares

A 68-by-68 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (157250)

How to Create a 68x68 Magic Square
with a Magic Sum of 157250

Step One: Draw a 68x68 blank grid and
shade in the two main diagonals

Step Three: Count the squares
(writing only on the shaded squares)

Step Four: Count the squares again, counting backward,
writing into the blank squares