A 27x27 Magic Square

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Other Magic Square Sizes: 3 * 5 * 7 * 9 * 11 * 13 * 15 * 17 * 19 *
21 * 23 * 25 * 27 * 29 * 31 * 33 * 35 * 37 * 39 * 41 * 43 * 45 * 47 * 49 * 51 * 53 * 55 * 57 * 59 *
61 * 63 * 65 * 67 * 69 * 71 * 73 * 75 * 77 * 79 * 81 * 83 * 85 * 87 * 89 * 91 * 93 * 95 * 97 * 99 *
A 27x27 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (9855)

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 27 by 27 magic square shown below has a magic sum equal to 9855.
Though magic squares can be made with non-consecutive and non-regular sequences, they are usually seen made up of consecutive numbers. To the best of my knowledge, the middle number in the sequence must be in the center square, and the magic sum will be equal to this middle number times the number of rows in the square. (In the case of 3x3 squares, I have proved that the middle number MUST be equal to one-third of the sum. See proof here.)
All odd-size Magic Squares can be made with the method shown below, which can be summarized as "start in the middle of the top and keep moving up and to the right except when you get blocked, in which case drop down and continue." The more detailed directions about creating these squares can be found further below. There are other methods for creating magic squares, but this is the easiest to learn. (There are basically only two methods of creating a 3x3 square but the larger squares have a large number of irregular variants, and probably other regular methods.)
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.

A 27-by-27 Magic Square

(Magic Sum = 9855)

Books about
Magic Squares
at Amazon.co.UK

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380	409	438	467	496	525	554	583	612	641	670	699	728	1	30	59	88	117	146	175	204	233	262	291	320	349	378
408	437	466	495	524	553	582	611	640	669	698	727	27	29	58	87	116	145	174	203	232	261	290	319	348	377	379
436	465	494	523	552	581	610	639	668	697	726	26	28	57	86	115	144	173	202	231	260	289	318	347	376	405	407
464	493	522	551	580	609	638	667	696	725	25	54	56	85	114	143	172	201	230	259	288	317	346	375	404	406	435
492	521	550	579	608	637	666	695	724	24	53	55	84	113	142	171	200	229	258	287	316	345	374	403	432	434	463
520	549	578	607	636	665	694	723	23	52	81	83	112	141	170	199	228	257	286	315	344	373	402	431	433	462	491
548	577	606	635	664	693	722	22	51	80	82	111	140	169	198	227	256	285	314	343	372	401	430	459	461	490	519
576	605	634	663	692	721	21	50	79	108	110	139	168	197	226	255	284	313	342	371	400	429	458	460	489	518	547
604	633	662	691	720	20	49	78	107	109	138	167	196	225	254	283	312	341	370	399	428	457	486	488	517	546	575
632	661	690	719	19	48	77	106	135	137	166	195	224	253	282	311	340	369	398	427	456	485	487	516	545	574	603
660	689	718	18	47	76	105	134	136	165	194	223	252	281	310	339	368	397	426	455	484	513	515	544	573	602	631
688	717	17	46	75	104	133	162	164	193	222	251	280	309	338	367	396	425	454	483	512	514	543	572	601	630	659
716	16	45	74	103	132	161	163	192	221	250	279	308	337	366	395	424	453	482	511	540	542	571	600	629	658	687
15	44	73	102	131	160	189	191	220	249	278	307	336	365	394	423	452	481	510	539	541	570	599	628	657	686	715
43	72	101	130	159	188	190	219	248	277	306	335	364	393	422	451	480	509	538	567	569	598	627	656	685	714	14
71	100	129	158	187	216	218	247	276	305	334	363	392	421	450	479	508	537	566	568	597	626	655	684	713	13	42
99	128	157	186	215	217	246	275	304	333	362	391	420	449	478	507	536	565	594	596	625	654	683	712	12	41	70
127	156	185	214	243	245	274	303	332	361	390	419	448	477	506	535	564	593	595	624	653	682	711	11	40	69	98
155	184	213	242	244	273	302	331	360	389	418	447	476	505	534	563	592	621	623	652	681	710	10	39	68	97	126
183	212	241	270	272	301	330	359	388	417	446	475	504	533	562	591	620	622	651	680	709	9	38	67	96	125	154
211	240	269	271	300	329	358	387	416	445	474	503	532	561	590	619	648	650	679	708	8	37	66	95	124	153	182
239	268	297	299	328	357	386	415	444	473	502	531	560	589	618	647	649	678	707	7	36	65	94	123	152	181	210
267	296	298	327	356	385	414	443	472	501	530	559	588	617	646	675	677	706	6	35	64	93	122	151	180	209	238
295	324	326	355	384	413	442	471	500	529	558	587	616	645	674	676	705	5	34	63	92	121	150	179	208	237	266
323	325	354	383	412	441	470	499	528	557	586	615	644	673	702	704	4	33	62	91	120	149	178	207	236	265	294
351	353	382	411	440	469	498	527	556	585	614	643	672	701	703	3	32	61	90	119	148	177	206	235	264	293	322
352	381	410	439	468	497	526	555	584	613	642	671	700	729	2	31	60	89	118	147	176	205	234	263	292	321	350

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How to Make Your Own Odd-Size Magic Square

Draw a square with any odd number of rows and columns.
Put your first number in the MIDDLE square of the TOP row.
Move UP and to the RIGHT to find the position for your next number, but ...
- if the next "position" is ABOVE a column, go to BOTTOM of that column (you can see a sample of this movement by looking at how we moved from 1 to 2 in the square above).
- if the next "position" is to the RIGHT of a row, go to LEFT of that row (you can see this movement between (14) and (15) above).
- if the next cell is already filled in, drop down to the cell below the number you just filled in for the next number (you can see this when you look at how we moved from 27 to the next number above).
- if the next cell is both ABOVE and to the RIGHT of the entire square, drop down to the cell below the number you just filled in for your next number (you can see this if you look at (378) and the next number).
Print your NEXT number into the new square you just moved to.
Repeat steps 3-4 until the entire square is full.
You can see how it works by printing this page and drawing arrows to see how it worked when I created this magic square

This Magic Square was created by John Knoderer, a webmaster and computer programmer who is available to telecommute to your location.

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.

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380	409	438	467	496	525	554	583	612	641	670	699	728	1	30	59	88	117	146	175	204	233	262	291	320	349	378
408	437	466	495	524	553	582	611	640	669	698	727	27	29	58	87	116	145	174	203	232	261	290	319	348	377	379
436	465	494	523	552	581	610	639	668	697	726	26	28	57	86	115	144	173	202	231	260	289	318	347	376	405	407
464	493	522	551	580	609	638	667	696	725	25	54	56	85	114	143	172	201	230	259	288	317	346	375	404	406	435
492	521	550	579	608	637	666	695	724	24	53	55	84	113	142	171	200	229	258	287	316	345	374	403	432	434	463
520	549	578	607	636	665	694	723	23	52	81	83	112	141	170	199	228	257	286	315	344	373	402	431	433	462	491
548	577	606	635	664	693	722	22	51	80	82	111	140	169	198	227	256	285	314	343	372	401	430	459	461	490	519
576	605	634	663	692	721	21	50	79	108	110	139	168	197	226	255	284	313	342	371	400	429	458	460	489	518	547
604	633	662	691	720	20	49	78	107	109	138	167	196	225	254	283	312	341	370	399	428	457	486	488	517	546	575
632	661	690	719	19	48	77	106	135	137	166	195	224	253	282	311	340	369	398	427	456	485	487	516	545	574	603
660	689	718	18	47	76	105	134	136	165	194	223	252	281	310	339	368	397	426	455	484	513	515	544	573	602	631
688	717	17	46	75	104	133	162	164	193	222	251	280	309	338	367	396	425	454	483	512	514	543	572	601	630	659
716	16	45	74	103	132	161	163	192	221	250	279	308	337	366	395	424	453	482	511	540	542	571	600	629	658	687
15	44	73	102	131	160	189	191	220	249	278	307	336	365	394	423	452	481	510	539	541	570	599	628	657	686	715
43	72	101	130	159	188	190	219	248	277	306	335	364	393	422	451	480	509	538	567	569	598	627	656	685	714	14
71	100	129	158	187	216	218	247	276	305	334	363	392	421	450	479	508	537	566	568	597	626	655	684	713	13	42
99	128	157	186	215	217	246	275	304	333	362	391	420	449	478	507	536	565	594	596	625	654	683	712	12	41	70
127	156	185	214	243	245	274	303	332	361	390	419	448	477	506	535	564	593	595	624	653	682	711	11	40	69	98
155	184	213	242	244	273	302	331	360	389	418	447	476	505	534	563	592	621	623	652	681	710	10	39	68	97	126
183	212	241	270	272	301	330	359	388	417	446	475	504	533	562	591	620	622	651	680	709	9	38	67	96	125	154
211	240	269	271	300	329	358	387	416	445	474	503	532	561	590	619	648	650	679	708	8	37	66	95	124	153	182
239	268	297	299	328	357	386	415	444	473	502	531	560	589	618	647	649	678	707	7	36	65	94	123	152	181	210
267	296	298	327	356	385	414	443	472	501	530	559	588	617	646	675	677	706	6	35	64	93	122	151	180	209	238
295	324	326	355	384	413	442	471	500	529	558	587	616	645	674	676	705	5	34	63	92	121	150	179	208	237	266
323	325	354	383	412	441	470	499	528	557	586	615	644	673	702	704	4	33	62	91	120	149	178	207	236	265	294
351	353	382	411	440	469	498	527	556	585	614	643	672	701	703	3	32	61	90	119	148	177	206	235	264	293	322
352	381	410	439	468	497	526	555	584	613	642	671	700	729	2	31	60	89	118	147	176	205	234	263	292	321	350

380	409	438	467	496	525	554	583	612	641	670	699	728	1	30	59	88	117	146	175	204	233	262	291	320	349	378
408	437	466	495	524	553	582	611	640	669	698	727	27	29	58	87	116	145	174	203	232	261	290	319	348	377	379
436	465	494	523	552	581	610	639	668	697	726	26	28	57	86	115	144	173	202	231	260	289	318	347	376	405	407
464	493	522	551	580	609	638	667	696	725	25	54	56	85	114	143	172	201	230	259	288	317	346	375	404	406	435
492	521	550	579	608	637	666	695	724	24	53	55	84	113	142	171	200	229	258	287	316	345	374	403	432	434	463
520	549	578	607	636	665	694	723	23	52	81	83	112	141	170	199	228	257	286	315	344	373	402	431	433	462	491
548	577	606	635	664	693	722	22	51	80	82	111	140	169	198	227	256	285	314	343	372	401	430	459	461	490	519
576	605	634	663	692	721	21	50	79	108	110	139	168	197	226	255	284	313	342	371	400	429	458	460	489	518	547
604	633	662	691	720	20	49	78	107	109	138	167	196	225	254	283	312	341	370	399	428	457	486	488	517	546	575
632	661	690	719	19	48	77	106	135	137	166	195	224	253	282	311	340	369	398	427	456	485	487	516	545	574	603
660	689	718	18	47	76	105	134	136	165	194	223	252	281	310	339	368	397	426	455	484	513	515	544	573	602	631
688	717	17	46	75	104	133	162	164	193	222	251	280	309	338	367	396	425	454	483	512	514	543	572	601	630	659
716	16	45	74	103	132	161	163	192	221	250	279	308	337	366	395	424	453	482	511	540	542	571	600	629	658	687
15	44	73	102	131	160	189	191	220	249	278	307	336	365	394	423	452	481	510	539	541	570	599	628	657	686	715
43	72	101	130	159	188	190	219	248	277	306	335	364	393	422	451	480	509	538	567	569	598	627	656	685	714	14
71	100	129	158	187	216	218	247	276	305	334	363	392	421	450	479	508	537	566	568	597	626	655	684	713	13	42
99	128	157	186	215	217	246	275	304	333	362	391	420	449	478	507	536	565	594	596	625	654	683	712	12	41	70
127	156	185	214	243	245	274	303	332	361	390	419	448	477	506	535	564	593	595	624	653	682	711	11	40	69	98
155	184	213	242	244	273	302	331	360	389	418	447	476	505	534	563	592	621	623	652	681	710	10	39	68	97	126
183	212	241	270	272	301	330	359	388	417	446	475	504	533	562	591	620	622	651	680	709	9	38	67	96	125	154
211	240	269	271	300	329	358	387	416	445	474	503	532	561	590	619	648	650	679	708	8	37	66	95	124	153	182
239	268	297	299	328	357	386	415	444	473	502	531	560	589	618	647	649	678	707	7	36	65	94	123	152	181	210
267	296	298	327	356	385	414	443	472	501	530	559	588	617	646	675	677	706	6	35	64	93	122	151	180	209	238
295	324	326	355	384	413	442	471	500	529	558	587	616	645	674	676	705	5	34	63	92	121	150	179	208	237	266
323	325	354	383	412	441	470	499	528	557	586	615	644	673	702	704	4	33	62	91	120	149	178	207	236	265	294
351	353	382	411	440	469	498	527	556	585	614	643	672	701	703	3	32	61	90	119	148	177	206	235	264	293	322
352	381	410	439	468	497	526	555	584	613	642	671	700	729	2	31	60	89	118	147	176	205	234	263	292	321	350

A 27x27 Magic Square

Scroll down to see a magic squarein which all rows, columns, and both diagonalsadd up to the same magic sum (9855)

Read more about Magic Squares

A 27-by-27 Magic Square

How to Make Your Own Odd-Size Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (9855)

380	409	438	467	496	525	554	583	612	641	670	699	728	1	30	59	88	117	146	175	204	233	262	291	320	349	378
408	437	466	495	524	553	582	611	640	669	698	727	27	29	58	87	116	145	174	203	232	261	290	319	348	377	379
436	465	494	523	552	581	610	639	668	697	726	26	28	57	86	115	144	173	202	231	260	289	318	347	376	405	407
464	493	522	551	580	609	638	667	696	725	25	54	56	85	114	143	172	201	230	259	288	317	346	375	404	406	435
492	521	550	579	608	637	666	695	724	24	53	55	84	113	142	171	200	229	258	287	316	345	374	403	432	434	463
520	549	578	607	636	665	694	723	23	52	81	83	112	141	170	199	228	257	286	315	344	373	402	431	433	462	491
548	577	606	635	664	693	722	22	51	80	82	111	140	169	198	227	256	285	314	343	372	401	430	459	461	490	519
576	605	634	663	692	721	21	50	79	108	110	139	168	197	226	255	284	313	342	371	400	429	458	460	489	518	547
604	633	662	691	720	20	49	78	107	109	138	167	196	225	254	283	312	341	370	399	428	457	486	488	517	546	575
632	661	690	719	19	48	77	106	135	137	166	195	224	253	282	311	340	369	398	427	456	485	487	516	545	574	603
660	689	718	18	47	76	105	134	136	165	194	223	252	281	310	339	368	397	426	455	484	513	515	544	573	602	631
688	717	17	46	75	104	133	162	164	193	222	251	280	309	338	367	396	425	454	483	512	514	543	572	601	630	659
716	16	45	74	103	132	161	163	192	221	250	279	308	337	366	395	424	453	482	511	540	542	571	600	629	658	687
15	44	73	102	131	160	189	191	220	249	278	307	336	365	394	423	452	481	510	539	541	570	599	628	657	686	715
43	72	101	130	159	188	190	219	248	277	306	335	364	393	422	451	480	509	538	567	569	598	627	656	685	714	14
71	100	129	158	187	216	218	247	276	305	334	363	392	421	450	479	508	537	566	568	597	626	655	684	713	13	42
99	128	157	186	215	217	246	275	304	333	362	391	420	449	478	507	536	565	594	596	625	654	683	712	12	41	70
127	156	185	214	243	245	274	303	332	361	390	419	448	477	506	535	564	593	595	624	653	682	711	11	40	69	98
155	184	213	242	244	273	302	331	360	389	418	447	476	505	534	563	592	621	623	652	681	710	10	39	68	97	126
183	212	241	270	272	301	330	359	388	417	446	475	504	533	562	591	620	622	651	680	709	9	38	67	96	125	154
211	240	269	271	300	329	358	387	416	445	474	503	532	561	590	619	648	650	679	708	8	37	66	95	124	153	182
239	268	297	299	328	357	386	415	444	473	502	531	560	589	618	647	649	678	707	7	36	65	94	123	152	181	210
267	296	298	327	356	385	414	443	472	501	530	559	588	617	646	675	677	706	6	35	64	93	122	151	180	209	238
295	324	326	355	384	413	442	471	500	529	558	587	616	645	674	676	705	5	34	63	92	121	150	179	208	237	266
323	325	354	383	412	441	470	499	528	557	586	615	644	673	702	704	4	33	62	91	120	149	178	207	236	265	294
351	353	382	411	440	469	498	527	556	585	614	643	672	701	703	3	32	61	90	119	148	177	206	235	264	293	322
352	381	410	439	468	497	526	555	584	613	642	671	700	729	2	31	60	89	118	147	176	205	234	263	292	321	350