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A 40x40 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (32020)

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 40 by 40 magic square shown below has a magic sum equal to 32020.
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 40x40 Magic Square
with a Magic Sum of 32020

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

Steps:

Draw a 40x40 blank grid and color in the two main diagonals.
Also, color in the two main diagonals of each 4x4 box within this 40x40 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 1600 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 1600, but this time, only write down the numbers in the non-shaded squares.

Step One: Draw a 40x40 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 (1600).

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

This time, you can either

Start counting backward, starting with 1600 (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 1600 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 1600 is already written in that square), write down 2 and 3, then skip 4, etc.

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A 40-by-40 Magic Square

(Magic Sum = 32020)

1	1599	1598	4	5	1595	1594	8	9	1591	1590	12	13	1587	1586	16	17	1583	1582	20	21	1579	1578	24	25	1575	1574	28	29	1571	1570	32	33	1567	1566	36	37	1563	1562	40
1560	42	43	1557	1556	46	47	1553	1552	50	51	1549	1548	54	55	1545	1544	58	59	1541	1540	62	63	1537	1536	66	67	1533	1532	70	71	1529	1528	74	75	1525	1524	78	79	1521
1520	82	83	1517	1516	86	87	1513	1512	90	91	1509	1508	94	95	1505	1504	98	99	1501	1500	102	103	1497	1496	106	107	1493	1492	110	111	1489	1488	114	115	1485	1484	118	119	1481
121	1479	1478	124	125	1475	1474	128	129	1471	1470	132	133	1467	1466	136	137	1463	1462	140	141	1459	1458	144	145	1455	1454	148	149	1451	1450	152	153	1447	1446	156	157	1443	1442	160
161	1439	1438	164	165	1435	1434	168	169	1431	1430	172	173	1427	1426	176	177	1423	1422	180	181	1419	1418	184	185	1415	1414	188	189	1411	1410	192	193	1407	1406	196	197	1403	1402	200
1400	202	203	1397	1396	206	207	1393	1392	210	211	1389	1388	214	215	1385	1384	218	219	1381	1380	222	223	1377	1376	226	227	1373	1372	230	231	1369	1368	234	235	1365	1364	238	239	1361
1360	242	243	1357	1356	246	247	1353	1352	250	251	1349	1348	254	255	1345	1344	258	259	1341	1340	262	263	1337	1336	266	267	1333	1332	270	271	1329	1328	274	275	1325	1324	278	279	1321
281	1319	1318	284	285	1315	1314	288	289	1311	1310	292	293	1307	1306	296	297	1303	1302	300	301	1299	1298	304	305	1295	1294	308	309	1291	1290	312	313	1287	1286	316	317	1283	1282	320
321	1279	1278	324	325	1275	1274	328	329	1271	1270	332	333	1267	1266	336	337	1263	1262	340	341	1259	1258	344	345	1255	1254	348	349	1251	1250	352	353	1247	1246	356	357	1243	1242	360
1240	362	363	1237	1236	366	367	1233	1232	370	371	1229	1228	374	375	1225	1224	378	379	1221	1220	382	383	1217	1216	386	387	1213	1212	390	391	1209	1208	394	395	1205	1204	398	399	1201
1200	402	403	1197	1196	406	407	1193	1192	410	411	1189	1188	414	415	1185	1184	418	419	1181	1180	422	423	1177	1176	426	427	1173	1172	430	431	1169	1168	434	435	1165	1164	438	439	1161
441	1159	1158	444	445	1155	1154	448	449	1151	1150	452	453	1147	1146	456	457	1143	1142	460	461	1139	1138	464	465	1135	1134	468	469	1131	1130	472	473	1127	1126	476	477	1123	1122	480
481	1119	1118	484	485	1115	1114	488	489	1111	1110	492	493	1107	1106	496	497	1103	1102	500	501	1099	1098	504	505	1095	1094	508	509	1091	1090	512	513	1087	1086	516	517	1083	1082	520
1080	522	523	1077	1076	526	527	1073	1072	530	531	1069	1068	534	535	1065	1064	538	539	1061	1060	542	543	1057	1056	546	547	1053	1052	550	551	1049	1048	554	555	1045	1044	558	559	1041
1040	562	563	1037	1036	566	567	1033	1032	570	571	1029	1028	574	575	1025	1024	578	579	1021	1020	582	583	1017	1016	586	587	1013	1012	590	591	1009	1008	594	595	1005	1004	598	599	1001
601	999	998	604	605	995	994	608	609	991	990	612	613	987	986	616	617	983	982	620	621	979	978	624	625	975	974	628	629	971	970	632	633	967	966	636	637	963	962	640
641	959	958	644	645	955	954	648	649	951	950	652	653	947	946	656	657	943	942	660	661	939	938	664	665	935	934	668	669	931	930	672	673	927	926	676	677	923	922	680
920	682	683	917	916	686	687	913	912	690	691	909	908	694	695	905	904	698	699	901	900	702	703	897	896	706	707	893	892	710	711	889	888	714	715	885	884	718	719	881
880	722	723	877	876	726	727	873	872	730	731	869	868	734	735	865	864	738	739	861	860	742	743	857	856	746	747	853	852	750	751	849	848	754	755	845	844	758	759	841
761	839	838	764	765	835	834	768	769	831	830	772	773	827	826	776	777	823	822	780	781	819	818	784	785	815	814	788	789	811	810	792	793	807	806	796	797	803	802	800
801	799	798	804	805	795	794	808	809	791	790	812	813	787	786	816	817	783	782	820	821	779	778	824	825	775	774	828	829	771	770	832	833	767	766	836	837	763	762	840
760	842	843	757	756	846	847	753	752	850	851	749	748	854	855	745	744	858	859	741	740	862	863	737	736	866	867	733	732	870	871	729	728	874	875	725	724	878	879	721
720	882	883	717	716	886	887	713	712	890	891	709	708	894	895	705	704	898	899	701	700	902	903	697	696	906	907	693	692	910	911	689	688	914	915	685	684	918	919	681
921	679	678	924	925	675	674	928	929	671	670	932	933	667	666	936	937	663	662	940	941	659	658	944	945	655	654	948	949	651	650	952	953	647	646	956	957	643	642	960
961	639	638	964	965	635	634	968	969	631	630	972	973	627	626	976	977	623	622	980	981	619	618	984	985	615	614	988	989	611	610	992	993	607	606	996	997	603	602	1000
600	1002	1003	597	596	1006	1007	593	592	1010	1011	589	588	1014	1015	585	584	1018	1019	581	580	1022	1023	577	576	1026	1027	573	572	1030	1031	569	568	1034	1035	565	564	1038	1039	561
560	1042	1043	557	556	1046	1047	553	552	1050	1051	549	548	1054	1055	545	544	1058	1059	541	540	1062	1063	537	536	1066	1067	533	532	1070	1071	529	528	1074	1075	525	524	1078	1079	521
1081	519	518	1084	1085	515	514	1088	1089	511	510	1092	1093	507	506	1096	1097	503	502	1100	1101	499	498	1104	1105	495	494	1108	1109	491	490	1112	1113	487	486	1116	1117	483	482	1120
1121	479	478	1124	1125	475	474	1128	1129	471	470	1132	1133	467	466	1136	1137	463	462	1140	1141	459	458	1144	1145	455	454	1148	1149	451	450	1152	1153	447	446	1156	1157	443	442	1160
440	1162	1163	437	436	1166	1167	433	432	1170	1171	429	428	1174	1175	425	424	1178	1179	421	420	1182	1183	417	416	1186	1187	413	412	1190	1191	409	408	1194	1195	405	404	1198	1199	401
400	1202	1203	397	396	1206	1207	393	392	1210	1211	389	388	1214	1215	385	384	1218	1219	381	380	1222	1223	377	376	1226	1227	373	372	1230	1231	369	368	1234	1235	365	364	1238	1239	361
1241	359	358	1244	1245	355	354	1248	1249	351	350	1252	1253	347	346	1256	1257	343	342	1260	1261	339	338	1264	1265	335	334	1268	1269	331	330	1272	1273	327	326	1276	1277	323	322	1280
1281	319	318	1284	1285	315	314	1288	1289	311	310	1292	1293	307	306	1296	1297	303	302	1300	1301	299	298	1304	1305	295	294	1308	1309	291	290	1312	1313	287	286	1316	1317	283	282	1320
280	1322	1323	277	276	1326	1327	273	272	1330	1331	269	268	1334	1335	265	264	1338	1339	261	260	1342	1343	257	256	1346	1347	253	252	1350	1351	249	248	1354	1355	245	244	1358	1359	241
240	1362	1363	237	236	1366	1367	233	232	1370	1371	229	228	1374	1375	225	224	1378	1379	221	220	1382	1383	217	216	1386	1387	213	212	1390	1391	209	208	1394	1395	205	204	1398	1399	201
1401	199	198	1404	1405	195	194	1408	1409	191	190	1412	1413	187	186	1416	1417	183	182	1420	1421	179	178	1424	1425	175	174	1428	1429	171	170	1432	1433	167	166	1436	1437	163	162	1440
1441	159	158	1444	1445	155	154	1448	1449	151	150	1452	1453	147	146	1456	1457	143	142	1460	1461	139	138	1464	1465	135	134	1468	1469	131	130	1472	1473	127	126	1476	1477	123	122	1480
120	1482	1483	117	116	1486	1487	113	112	1490	1491	109	108	1494	1495	105	104	1498	1499	101	100	1502	1503	97	96	1506	1507	93	92	1510	1511	89	88	1514	1515	85	84	1518	1519	81
80	1522	1523	77	76	1526	1527	73	72	1530	1531	69	68	1534	1535	65	64	1538	1539	61	60	1542	1543	57	56	1546	1547	53	52	1550	1551	49	48	1554	1555	45	44	1558	1559	41
1561	39	38	1564	1565	35	34	1568	1569	31	30	1572	1573	27	26	1576	1577	23	22	1580	1581	19	18	1584	1585	15	14	1588	1589	11	10	1592	1593	7	6	1596	1597	3	2	1600

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A 40x40 Magic Square

Scroll down to see a magic square in which all rows, columns, and both diagonals add up to the same magic sum (32020)

Read more about Magic Squares

How to Create a 40x40 Magic Square with a Magic Sum of 32020

Step One: Draw a 40x40 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 40-by-40 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (32020)

How to Create a 40x40 Magic Square
with a Magic Sum of 32020

Step One: Draw a 40x40 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