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Other 4N+0 Magic Square Sizes:
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A 64x64 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (131104)

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 64 by 64 magic square shown below has a magic sum equal to 131104.
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 64x64 Magic Square
with a Magic Sum of 131104

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

Steps:

Draw a 64x64 blank grid and color in the two main diagonals.
Also, color in the two main diagonals of each 4x4 box within this 64x64 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 4096 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 4096, but this time, only write down the numbers in the non-shaded squares.

Step One: Draw a 64x64 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 (4096).

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

This time, you can either

Start counting backward, starting with 4096 (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 4096 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 4096 is already written in that square), write down 2 and 3, then skip 4, etc.

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A 64-by-64 Magic Square

(Magic Sum = 131104)

1	4095	4094	4	5	4091	4090	8	9	4087	4086	12	13	4083	4082	16	17	4079	4078	20	21	4075	4074	24	25	4071	4070	28	29	4067	4066	32	33	4063	4062	36	37	4059	4058	40	41	4055	4054	44	45	4051	4050	48	49	4047	4046	52	53	4043	4042	56	57	4039	4038	60	61	4035	4034	64
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2305	1791	1790	2308	2309	1787	1786	2312	2313	1783	1782	2316	2317	1779	1778	2320	2321	1775	1774	2324	2325	1771	1770	2328	2329	1767	1766	2332	2333	1763	1762	2336	2337	1759	1758	2340	2341	1755	1754	2344	2345	1751	1750	2348	2349	1747	1746	2352	2353	1743	1742	2356	2357	1739	1738	2360	2361	1735	1734	2364	2365	1731	1730	2368
1728	2370	2371	1725	1724	2374	2375	1721	1720	2378	2379	1717	1716	2382	2383	1713	1712	2386	2387	1709	1708	2390	2391	1705	1704	2394	2395	1701	1700	2398	2399	1697	1696	2402	2403	1693	1692	2406	2407	1689	1688	2410	2411	1685	1684	2414	2415	1681	1680	2418	2419	1677	1676	2422	2423	1673	1672	2426	2427	1669	1668	2430	2431	1665
1664	2434	2435	1661	1660	2438	2439	1657	1656	2442	2443	1653	1652	2446	2447	1649	1648	2450	2451	1645	1644	2454	2455	1641	1640	2458	2459	1637	1636	2462	2463	1633	1632	2466	2467	1629	1628	2470	2471	1625	1624	2474	2475	1621	1620	2478	2479	1617	1616	2482	2483	1613	1612	2486	2487	1609	1608	2490	2491	1605	1604	2494	2495	1601
2497	1599	1598	2500	2501	1595	1594	2504	2505	1591	1590	2508	2509	1587	1586	2512	2513	1583	1582	2516	2517	1579	1578	2520	2521	1575	1574	2524	2525	1571	1570	2528	2529	1567	1566	2532	2533	1563	1562	2536	2537	1559	1558	2540	2541	1555	1554	2544	2545	1551	1550	2548	2549	1547	1546	2552	2553	1543	1542	2556	2557	1539	1538	2560
2561	1535	1534	2564	2565	1531	1530	2568	2569	1527	1526	2572	2573	1523	1522	2576	2577	1519	1518	2580	2581	1515	1514	2584	2585	1511	1510	2588	2589	1507	1506	2592	2593	1503	1502	2596	2597	1499	1498	2600	2601	1495	1494	2604	2605	1491	1490	2608	2609	1487	1486	2612	2613	1483	1482	2616	2617	1479	1478	2620	2621	1475	1474	2624
1472	2626	2627	1469	1468	2630	2631	1465	1464	2634	2635	1461	1460	2638	2639	1457	1456	2642	2643	1453	1452	2646	2647	1449	1448	2650	2651	1445	1444	2654	2655	1441	1440	2658	2659	1437	1436	2662	2663	1433	1432	2666	2667	1429	1428	2670	2671	1425	1424	2674	2675	1421	1420	2678	2679	1417	1416	2682	2683	1413	1412	2686	2687	1409
1408	2690	2691	1405	1404	2694	2695	1401	1400	2698	2699	1397	1396	2702	2703	1393	1392	2706	2707	1389	1388	2710	2711	1385	1384	2714	2715	1381	1380	2718	2719	1377	1376	2722	2723	1373	1372	2726	2727	1369	1368	2730	2731	1365	1364	2734	2735	1361	1360	2738	2739	1357	1356	2742	2743	1353	1352	2746	2747	1349	1348	2750	2751	1345
2753	1343	1342	2756	2757	1339	1338	2760	2761	1335	1334	2764	2765	1331	1330	2768	2769	1327	1326	2772	2773	1323	1322	2776	2777	1319	1318	2780	2781	1315	1314	2784	2785	1311	1310	2788	2789	1307	1306	2792	2793	1303	1302	2796	2797	1299	1298	2800	2801	1295	1294	2804	2805	1291	1290	2808	2809	1287	1286	2812	2813	1283	1282	2816
2817	1279	1278	2820	2821	1275	1274	2824	2825	1271	1270	2828	2829	1267	1266	2832	2833	1263	1262	2836	2837	1259	1258	2840	2841	1255	1254	2844	2845	1251	1250	2848	2849	1247	1246	2852	2853	1243	1242	2856	2857	1239	1238	2860	2861	1235	1234	2864	2865	1231	1230	2868	2869	1227	1226	2872	2873	1223	1222	2876	2877	1219	1218	2880
1216	2882	2883	1213	1212	2886	2887	1209	1208	2890	2891	1205	1204	2894	2895	1201	1200	2898	2899	1197	1196	2902	2903	1193	1192	2906	2907	1189	1188	2910	2911	1185	1184	2914	2915	1181	1180	2918	2919	1177	1176	2922	2923	1173	1172	2926	2927	1169	1168	2930	2931	1165	1164	2934	2935	1161	1160	2938	2939	1157	1156	2942	2943	1153
1152	2946	2947	1149	1148	2950	2951	1145	1144	2954	2955	1141	1140	2958	2959	1137	1136	2962	2963	1133	1132	2966	2967	1129	1128	2970	2971	1125	1124	2974	2975	1121	1120	2978	2979	1117	1116	2982	2983	1113	1112	2986	2987	1109	1108	2990	2991	1105	1104	2994	2995	1101	1100	2998	2999	1097	1096	3002	3003	1093	1092	3006	3007	1089
3009	1087	1086	3012	3013	1083	1082	3016	3017	1079	1078	3020	3021	1075	1074	3024	3025	1071	1070	3028	3029	1067	1066	3032	3033	1063	1062	3036	3037	1059	1058	3040	3041	1055	1054	3044	3045	1051	1050	3048	3049	1047	1046	3052	3053	1043	1042	3056	3057	1039	1038	3060	3061	1035	1034	3064	3065	1031	1030	3068	3069	1027	1026	3072
3073	1023	1022	3076	3077	1019	1018	3080	3081	1015	1014	3084	3085	1011	1010	3088	3089	1007	1006	3092	3093	1003	1002	3096	3097	999	998	3100	3101	995	994	3104	3105	991	990	3108	3109	987	986	3112	3113	983	982	3116	3117	979	978	3120	3121	975	974	3124	3125	971	970	3128	3129	967	966	3132	3133	963	962	3136
960	3138	3139	957	956	3142	3143	953	952	3146	3147	949	948	3150	3151	945	944	3154	3155	941	940	3158	3159	937	936	3162	3163	933	932	3166	3167	929	928	3170	3171	925	924	3174	3175	921	920	3178	3179	917	916	3182	3183	913	912	3186	3187	909	908	3190	3191	905	904	3194	3195	901	900	3198	3199	897
896	3202	3203	893	892	3206	3207	889	888	3210	3211	885	884	3214	3215	881	880	3218	3219	877	876	3222	3223	873	872	3226	3227	869	868	3230	3231	865	864	3234	3235	861	860	3238	3239	857	856	3242	3243	853	852	3246	3247	849	848	3250	3251	845	844	3254	3255	841	840	3258	3259	837	836	3262	3263	833
3265	831	830	3268	3269	827	826	3272	3273	823	822	3276	3277	819	818	3280	3281	815	814	3284	3285	811	810	3288	3289	807	806	3292	3293	803	802	3296	3297	799	798	3300	3301	795	794	3304	3305	791	790	3308	3309	787	786	3312	3313	783	782	3316	3317	779	778	3320	3321	775	774	3324	3325	771	770	3328
3329	767	766	3332	3333	763	762	3336	3337	759	758	3340	3341	755	754	3344	3345	751	750	3348	3349	747	746	3352	3353	743	742	3356	3357	739	738	3360	3361	735	734	3364	3365	731	730	3368	3369	727	726	3372	3373	723	722	3376	3377	719	718	3380	3381	715	714	3384	3385	711	710	3388	3389	707	706	3392
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640	3458	3459	637	636	3462	3463	633	632	3466	3467	629	628	3470	3471	625	624	3474	3475	621	620	3478	3479	617	616	3482	3483	613	612	3486	3487	609	608	3490	3491	605	604	3494	3495	601	600	3498	3499	597	596	3502	3503	593	592	3506	3507	589	588	3510	3511	585	584	3514	3515	581	580	3518	3519	577
3521	575	574	3524	3525	571	570	3528	3529	567	566	3532	3533	563	562	3536	3537	559	558	3540	3541	555	554	3544	3545	551	550	3548	3549	547	546	3552	3553	543	542	3556	3557	539	538	3560	3561	535	534	3564	3565	531	530	3568	3569	527	526	3572	3573	523	522	3576	3577	519	518	3580	3581	515	514	3584
3585	511	510	3588	3589	507	506	3592	3593	503	502	3596	3597	499	498	3600	3601	495	494	3604	3605	491	490	3608	3609	487	486	3612	3613	483	482	3616	3617	479	478	3620	3621	475	474	3624	3625	471	470	3628	3629	467	466	3632	3633	463	462	3636	3637	459	458	3640	3641	455	454	3644	3645	451	450	3648
448	3650	3651	445	444	3654	3655	441	440	3658	3659	437	436	3662	3663	433	432	3666	3667	429	428	3670	3671	425	424	3674	3675	421	420	3678	3679	417	416	3682	3683	413	412	3686	3687	409	408	3690	3691	405	404	3694	3695	401	400	3698	3699	397	396	3702	3703	393	392	3706	3707	389	388	3710	3711	385
384	3714	3715	381	380	3718	3719	377	376	3722	3723	373	372	3726	3727	369	368	3730	3731	365	364	3734	3735	361	360	3738	3739	357	356	3742	3743	353	352	3746	3747	349	348	3750	3751	345	344	3754	3755	341	340	3758	3759	337	336	3762	3763	333	332	3766	3767	329	328	3770	3771	325	324	3774	3775	321
3777	319	318	3780	3781	315	314	3784	3785	311	310	3788	3789	307	306	3792	3793	303	302	3796	3797	299	298	3800	3801	295	294	3804	3805	291	290	3808	3809	287	286	3812	3813	283	282	3816	3817	279	278	3820	3821	275	274	3824	3825	271	270	3828	3829	267	266	3832	3833	263	262	3836	3837	259	258	3840
3841	255	254	3844	3845	251	250	3848	3849	247	246	3852	3853	243	242	3856	3857	239	238	3860	3861	235	234	3864	3865	231	230	3868	3869	227	226	3872	3873	223	222	3876	3877	219	218	3880	3881	215	214	3884	3885	211	210	3888	3889	207	206	3892	3893	203	202	3896	3897	199	198	3900	3901	195	194	3904
192	3906	3907	189	188	3910	3911	185	184	3914	3915	181	180	3918	3919	177	176	3922	3923	173	172	3926	3927	169	168	3930	3931	165	164	3934	3935	161	160	3938	3939	157	156	3942	3943	153	152	3946	3947	149	148	3950	3951	145	144	3954	3955	141	140	3958	3959	137	136	3962	3963	133	132	3966	3967	129
128	3970	3971	125	124	3974	3975	121	120	3978	3979	117	116	3982	3983	113	112	3986	3987	109	108	3990	3991	105	104	3994	3995	101	100	3998	3999	97	96	4002	4003	93	92	4006	4007	89	88	4010	4011	85	84	4014	4015	81	80	4018	4019	77	76	4022	4023	73	72	4026	4027	69	68	4030	4031	65
4033	63	62	4036	4037	59	58	4040	4041	55	54	4044	4045	51	50	4048	4049	47	46	4052	4053	43	42	4056	4057	39	38	4060	4061	35	34	4064	4065	31	30	4068	4069	27	26	4072	4073	23	22	4076	4077	19	18	4080	4081	15	14	4084	4085	11	10	4088	4089	7	6	4092	4093	3	2	4096

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 64x64 Magic Square

Scroll down to see a magic square in which all rows, columns, and both diagonals add up to the same magic sum (131104)

Read more about Magic Squares

How to Create a 64x64 Magic Square with a Magic Sum of 131104

Step One: Draw a 64x64 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 64-by-64 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (131104)

How to Create a 64x64 Magic Square
with a Magic Sum of 131104

Step One: Draw a 64x64 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