Books about
Magic Squares
at Amazon.Canada

Amazing Math Whiz
products available:

Buttons, Magnets,
t-shirts, Tote bags,
Teddy bears, & more
Books about Magic Squares at Amazon.com

Be sure to visit the
Magic Squares Main Menu
for other topics including Irregular Squares, Algebraic Squares and more.

Other 4N+0 Magic Square Sizes:
Detailed Directions for: 4x4 * 8x8 * 12x12 *
and many more examples
* 16 * 20 * 24 * 28 * 32 * 36 * 40 * 44 * 48 * 52 * 56 *
* 60 * 64 * 68 * 72 * 76 * 80 * 84 * 88 * 92 * 96 *

A 36x36 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (23346)

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 36 by 36 magic square shown below has a magic sum equal to 23346.
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.

Read more about Magic Squares
If you'd like to read more about magic squares, click one of these links to look for books about Magic Squares at Amazon.com, at Amazon.canada, and at Amazon.co.UK.

Books about
Magic Squares
at Amazon.co.UK

Support Our Site
by purchasing our
Amazing Math Whiz
products

and other
Amazing Messages

How to Create a 36x36 Magic Square
with a Magic Sum of 23346

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

Steps:

Draw a 36x36 blank grid and color in the two main diagonals.
Also, color in the two main diagonals of each 4x4 box within this 36x36 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 1296 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 1296, but this time, only write down the numbers in the non-shaded squares.

Step One: Draw a 36x36 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 (1296).

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

This time, you can either

Start counting backward, starting with 1296 (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 1296 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 1296 is already written in that square), write down 2 and 3, then skip 4, etc.

We have many
Amazing Message
stores.

Click each image to see
other amazing product choices

(each product imprinted
with an actual
solvable maze puzzle

Amazing Teacher

Amazing Student

Amazing Reader

Amazing Artist

See our choices @
www.mazes.com/cp

Please
Support Our Site

buy
Amazing Math Whiz
products from
our online store

Many other
Amazing Messages
are available

Please visit
www.MAZES.com/cp/
for details

A 36-by-36 Magic Square

(Magic Sum = 23346)

1	1295	1294	4	5	1291	1290	8	9	1287	1286	12	13	1283	1282	16	17	1279	1278	20	21	1275	1274	24	25	1271	1270	28	29	1267	1266	32	33	1263	1262	36
1260	38	39	1257	1256	42	43	1253	1252	46	47	1249	1248	50	51	1245	1244	54	55	1241	1240	58	59	1237	1236	62	63	1233	1232	66	67	1229	1228	70	71	1225
1224	74	75	1221	1220	78	79	1217	1216	82	83	1213	1212	86	87	1209	1208	90	91	1205	1204	94	95	1201	1200	98	99	1197	1196	102	103	1193	1192	106	107	1189
109	1187	1186	112	113	1183	1182	116	117	1179	1178	120	121	1175	1174	124	125	1171	1170	128	129	1167	1166	132	133	1163	1162	136	137	1159	1158	140	141	1155	1154	144
145	1151	1150	148	149	1147	1146	152	153	1143	1142	156	157	1139	1138	160	161	1135	1134	164	165	1131	1130	168	169	1127	1126	172	173	1123	1122	176	177	1119	1118	180
1116	182	183	1113	1112	186	187	1109	1108	190	191	1105	1104	194	195	1101	1100	198	199	1097	1096	202	203	1093	1092	206	207	1089	1088	210	211	1085	1084	214	215	1081
1080	218	219	1077	1076	222	223	1073	1072	226	227	1069	1068	230	231	1065	1064	234	235	1061	1060	238	239	1057	1056	242	243	1053	1052	246	247	1049	1048	250	251	1045
253	1043	1042	256	257	1039	1038	260	261	1035	1034	264	265	1031	1030	268	269	1027	1026	272	273	1023	1022	276	277	1019	1018	280	281	1015	1014	284	285	1011	1010	288
289	1007	1006	292	293	1003	1002	296	297	999	998	300	301	995	994	304	305	991	990	308	309	987	986	312	313	983	982	316	317	979	978	320	321	975	974	324
972	326	327	969	968	330	331	965	964	334	335	961	960	338	339	957	956	342	343	953	952	346	347	949	948	350	351	945	944	354	355	941	940	358	359	937
936	362	363	933	932	366	367	929	928	370	371	925	924	374	375	921	920	378	379	917	916	382	383	913	912	386	387	909	908	390	391	905	904	394	395	901
397	899	898	400	401	895	894	404	405	891	890	408	409	887	886	412	413	883	882	416	417	879	878	420	421	875	874	424	425	871	870	428	429	867	866	432
433	863	862	436	437	859	858	440	441	855	854	444	445	851	850	448	449	847	846	452	453	843	842	456	457	839	838	460	461	835	834	464	465	831	830	468
828	470	471	825	824	474	475	821	820	478	479	817	816	482	483	813	812	486	487	809	808	490	491	805	804	494	495	801	800	498	499	797	796	502	503	793
792	506	507	789	788	510	511	785	784	514	515	781	780	518	519	777	776	522	523	773	772	526	527	769	768	530	531	765	764	534	535	761	760	538	539	757
541	755	754	544	545	751	750	548	549	747	746	552	553	743	742	556	557	739	738	560	561	735	734	564	565	731	730	568	569	727	726	572	573	723	722	576
577	719	718	580	581	715	714	584	585	711	710	588	589	707	706	592	593	703	702	596	597	699	698	600	601	695	694	604	605	691	690	608	609	687	686	612
684	614	615	681	680	618	619	677	676	622	623	673	672	626	627	669	668	630	631	665	664	634	635	661	660	638	639	657	656	642	643	653	652	646	647	649
648	650	651	645	644	654	655	641	640	658	659	637	636	662	663	633	632	666	667	629	628	670	671	625	624	674	675	621	620	678	679	617	616	682	683	613
685	611	610	688	689	607	606	692	693	603	602	696	697	599	598	700	701	595	594	704	705	591	590	708	709	587	586	712	713	583	582	716	717	579	578	720
721	575	574	724	725	571	570	728	729	567	566	732	733	563	562	736	737	559	558	740	741	555	554	744	745	551	550	748	749	547	546	752	753	543	542	756
540	758	759	537	536	762	763	533	532	766	767	529	528	770	771	525	524	774	775	521	520	778	779	517	516	782	783	513	512	786	787	509	508	790	791	505
504	794	795	501	500	798	799	497	496	802	803	493	492	806	807	489	488	810	811	485	484	814	815	481	480	818	819	477	476	822	823	473	472	826	827	469
829	467	466	832	833	463	462	836	837	459	458	840	841	455	454	844	845	451	450	848	849	447	446	852	853	443	442	856	857	439	438	860	861	435	434	864
865	431	430	868	869	427	426	872	873	423	422	876	877	419	418	880	881	415	414	884	885	411	410	888	889	407	406	892	893	403	402	896	897	399	398	900
396	902	903	393	392	906	907	389	388	910	911	385	384	914	915	381	380	918	919	377	376	922	923	373	372	926	927	369	368	930	931	365	364	934	935	361
360	938	939	357	356	942	943	353	352	946	947	349	348	950	951	345	344	954	955	341	340	958	959	337	336	962	963	333	332	966	967	329	328	970	971	325
973	323	322	976	977	319	318	980	981	315	314	984	985	311	310	988	989	307	306	992	993	303	302	996	997	299	298	1000	1001	295	294	1004	1005	291	290	1008
1009	287	286	1012	1013	283	282	1016	1017	279	278	1020	1021	275	274	1024	1025	271	270	1028	1029	267	266	1032	1033	263	262	1036	1037	259	258	1040	1041	255	254	1044
252	1046	1047	249	248	1050	1051	245	244	1054	1055	241	240	1058	1059	237	236	1062	1063	233	232	1066	1067	229	228	1070	1071	225	224	1074	1075	221	220	1078	1079	217
216	1082	1083	213	212	1086	1087	209	208	1090	1091	205	204	1094	1095	201	200	1098	1099	197	196	1102	1103	193	192	1106	1107	189	188	1110	1111	185	184	1114	1115	181
1117	179	178	1120	1121	175	174	1124	1125	171	170	1128	1129	167	166	1132	1133	163	162	1136	1137	159	158	1140	1141	155	154	1144	1145	151	150	1148	1149	147	146	1152
1153	143	142	1156	1157	139	138	1160	1161	135	134	1164	1165	131	130	1168	1169	127	126	1172	1173	123	122	1176	1177	119	118	1180	1181	115	114	1184	1185	111	110	1188
108	1190	1191	105	104	1194	1195	101	100	1198	1199	97	96	1202	1203	93	92	1206	1207	89	88	1210	1211	85	84	1214	1215	81	80	1218	1219	77	76	1222	1223	73
72	1226	1227	69	68	1230	1231	65	64	1234	1235	61	60	1238	1239	57	56	1242	1243	53	52	1246	1247	49	48	1250	1251	45	44	1254	1255	41	40	1258	1259	37
1261	35	34	1264	1265	31	30	1268	1269	27	26	1272	1273	23	22	1276	1277	19	18	1280	1281	15	14	1284	1285	11	10	1288	1289	7	6	1292	1293	3	2	1296

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

www.MAZES.com
www.GodLovesEveryone.org
www.DriverExam.org

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 36x36 Magic Square

Scroll down to see a magic square in which all rows, columns, and both diagonals add up to the same magic sum (23346)

Read more about Magic Squares

How to Create a 36x36 Magic Square with a Magic Sum of 23346

Step One: Draw a 36x36 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 36-by-36 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (23346)

How to Create a 36x36 Magic Square
with a Magic Sum of 23346

Step One: Draw a 36x36 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