Inlaid 22x22 magic square

The challenge was to construct an inlaid magic square with even and odd inlay magic squares in it. See below how I did it.

The design

The design of the 20x20 inlay square is as follows:

In the corners are four 7x7 panmagic squares. Around the 7x7 panmagic squares are half borders. The ‘cross’ in the middle consists of five panmagic 4x4 squares (and eight half panmagic suares, in which two times two numbers have been swapped to make the magic square valid; see explained later). In the 20x20 inlay square (at first) are the numbers 1 up to 400 (later on added by 42). I used the numbers 103 up to 298 to construct the four panmagic 7x7 squares. I used the numbers 73 up to 102 and 299 up to 328 to construct the half borders. I used the numbers 1 up to 72 and 329 up to 400 to construct the five whole and eight half panmagic 4x4 squares.

The four panmagic 7x7 squares

To construct the four panmagic 7x7 squares we use the same method to construct a composite 21x21 magic square. As row coordinates we use the numbers 0 up to 6. As column coordinates we use the numbers 0 up to 27 and we try to get the four panmagic 7x7 squares as proportional as possible.

Column coordinates first square Row coordinates first square

0	4	11	13	18	23	25	0	1	2	3	4	5	6
11	13	18	23	25	0	4	3	4	5	6	0	1	2
18	23	25	0	4	11	13	6	0	1	2	3	4	5
25	0	4	11	13	18	23	2	3	4	5	6	0	1
4	11	13	18	23	25	0	5	6	0	1	2	3	4
13	18	23	25	0	4	11	1	2	3	4	5	6	0
23	25	0	4	11	13	18	4	5	6	0	1	2	3

7x column+1x row coord.+1+102 = First panmagic 7x7 square

1	30	80	95	131	167	182	103	132	182	197	233	269	284
81	96	132	168	176	2	31	183	198	234	270	278	104	133
133	162	177	3	32	82	97	235	264	279	105	134	184	199
178	4	33	83	98	127	163	280	106	135	185	200	229	265
34	84	92	128	164	179	5	136	186	194	230	266	281	107
93	129	165	180	6	35	78	195	231	267	282	108	137	180
166	181	7	29	79	94	130	268	283	109	131	181	196	232

Column coordinates second square Row coordinates second square

2	5	9	15	16	21	26	0	1	2	3	4	5	6
9	15	16	21	26	2	5	3	4	5	6	0	1	2
16	21	26	2	5	9	15	6	0	1	2	3	4	5
26	2	5	9	15	16	21	2	3	4	5	6	0	1
5	9	15	16	21	26	2	5	6	0	1	2	3	4
15	16	21	26	2	5	9	1	2	3	4	5	6	0
21	26	2	5	9	15	16	4	5	6	0	1	2	3

7x column+1x row coord.+1+102 = Second panmagic 7x7 square

15	37	66	109	117	153	189	117	139	168	211	219	255	291
67	110	118	154	183	16	38	169	212	220	256	285	118	140
119	148	184	17	39	68	111	221	250	286	119	141	170	213
185	18	40	69	112	113	149	287	120	142	171	214	215	251
41	70	106	114	150	186	19	143	172	208	216	252	288	121
107	115	151	187	20	42	64	209	217	253	289	122	144	166
152	188	21	36	65	108	116	254	290	123	138	167	210	218

Column coordinates third square Row coordinates third square

3	7	10	14	17	20	24	0	1	2	3	4	5	6
10	14	17	20	24	3	7	3	4	5	6	0	1	2
17	20	24	3	7	10	14	6	0	1	2	3	4	5
24	3	7	10	14	17	20	2	3	4	5	6	0	1
7	10	14	17	20	24	3	5	6	0	1	2	3	4
14	17	20	24	3	7	10	1	2	3	4	5	6	0
20	24	3	7	10	14	17	4	5	6	0	1	2	3

7x column+1x row coord.+1+102 = Third panmagic 7x7 square

22	51	73	102	124	146	175	124	153	175	204	226	248	277
74	103	125	147	169	23	52	176	205	227	249	271	125	154
126	141	170	24	53	75	104	228	243	272	126	155	177	206
171	25	54	76	105	120	142	273	127	156	178	207	222	244
55	77	99	121	143	172	26	157	179	201	223	245	274	128
100	122	144	173	27	56	71	202	224	246	275	129	158	173
145	174	28	50	72	101	123	247	276	130	152	174	203	225

Column coordinates fourth square Row coordinates fourth square

1	6	8	12	19	22	27	0	1	2	3	4	5	6
8	12	19	22	27	1	6	3	4	5	6	0	1	2
19	22	27	1	6	8	12	6	0	1	2	3	4	5
27	1	6	8	12	19	22	2	3	4	5	6	0	1
6	8	12	19	22	27	1	5	6	0	1	2	3	4
12	19	22	27	1	6	8	1	2	3	4	5	6	0
22	27	1	6	8	12	19	4	5	6	0	1	2	3

7x column+1x row coord.+1+102 = Fourth panmagic 7x7 square

8	44	59	88	138	160	196	110	146	161	190	240	262	298
60	89	139	161	190	9	45	162	191	241	263	292	111	147
140	155	191	10	46	61	90	242	257	293	112	148	163	192
192	11	47	62	91	134	156	294	113	149	164	193	236	258
48	63	85	135	157	193	12	150	165	187	237	259	295	114
86	136	158	194	13	49	57	188	238	260	296	115	151	159
159	195	14	43	58	87	137	261	297	116	145	160	189	239

Put the first and second panmagic 7x7 square on top and the third and fourth panmagic 7x7 square at the bottom. That give the following sum totals of the rows, the columns and the diagonals:

		2807	2807	2807	2807	2807	2807	2807	2807	2807	2807	2807	2807	2807	2807
	2807															2807
2800		103	132	182	197	233	269	284	117	139	168	211	219	255	291
2800		183	198	234	270	278	104	133	169	212	220	256	285	118	140
2800		235	264	279	105	134	184	199	221	250	286	119	141	170	213
2800		280	106	135	185	200	229	265	287	120	142	171	214	215	251
2800		136	186	194	230	266	281	107	143	172	208	216	252	288	121
2800		195	231	267	282	108	137	180	209	217	253	289	122	144	166
2800		268	283	109	131	181	196	232	254	290	123	138	167	210	218






2814		124	153	175	204	226	248	277	110	146	161	190	240	262	298
2814		176	205	227	249	271	125	154	162	191	241	263	292	111	147
2814		228	243	272	126	155	177	206	242	257	293	112	148	163	192
2814		273	127	156	178	207	222	244	294	113	149	164	193	236	258
2814		157	179	201	223	245	274	128	150	165	187	237	259	295	114
2814		202	224	246	275	129	158	173	188	238	260	296	115	151	159
2814		247	276	130	152	174	203	225	261	297	116	145	160	189	239

Notify that op the sum of the rows is (14/20 x 4010 =) 2807 -/- 7 respectively +/+ 7. We correct the difference when we construct the half borders around the 7x7 magic squares.

The four half borders

To construct the vertical parts of the half borders excluding the corners we use the numbers 89 up to 102 and 299 up to 312. We combine the numbers to get the sum of 408 for the top vertical parts and the sum of 394 for the bottom vertical parts. To construct the horizontal parts including the corners we combine the remaining numbers, taking care that the sum is each time 401. With exception of the half borders, the sum of all rows, columns and diagonals is now 16/20 of the magic sum of 4010 that is 3208.

		3208	3208	3208	3208	3208	3208	3208	3216	3200	3208	3208	3208	3208	3208	3208	3208
	3208								+8	-8								3208
3208		103	132	182	197	233	269	284	312	96	117	139	168	211	219	255	291
3208		183	198	234	270	278	104	133	311	97	169	212	220	256	285	118	140
3208		235	264	279	105	134	184	199	310	98	221	250	286	119	141	170	213
3208		280	106	135	185	200	229	265	309	99	287	120	142	171	214	215	251
3208		136	186	194	230	266	281	107	308	100	143	172	208	216	252	288	121
3208		195	231	267	282	108	137	180	307	101	209	217	253	289	122	144	166
3208		268	283	109	131	181	196	232	306	102	254	290	123	138	167	210	218
3270	+62	328	327	326	325	324	323	322	88	80	87	86	85	84	83	82	320




3146	-62	73	74	75	76	77	78	79	321	313	314	315	316	317	318	319	81
3208		124	153	175	204	226	248	277	89	305	110	146	161	190	240	262	298
3208		176	205	227	249	271	125	154	90	304	162	191	241	263	292	111	147
3208		228	243	272	126	155	177	206	91	303	242	257	293	112	148	163	192
3208		273	127	156	178	207	222	244	92	302	294	113	149	164	193	236	258
3208		157	179	201	223	245	274	128	93	301	150	165	187	237	259	295	114
3208		202	224	246	275	129	158	173	94	300	188	238	260	296	115	151	159
3208		247	276	130	152	174	203	225	95	299	261	297	116	145	160	189	239

Notify that the sum of the two horizontal half borders is (16/20 x 4010 =) 3208 +/+ 62 respectively -/- 62 and the sum of the two vertical half borders is 3208 +/+ 8 respectively -/- 8. These differences will be corrected with the construction of the eight half panmagic 4x4 squares.

The (8x1/2 + 5x1 =) 9 panmagic 4x4 squares

To construct the 9 panmagic 4x4 squares we need the 72 lowest and the 72 highest numbers. We construct 9 proportional panmagic 4x4 squares (with the magic sum of 4/20 x 4010 = 802) by using the Khajuraho method.

7	396	1	398	15	388	9	390	23	380	17	382	31	372	25	374	39	364	33	366
2	397	8	395	10	389	16	387	18	381	24	379	26	373	32	371	34	365	40	363
400	3	394	5	392	11	386	13	384	19	378	21	376	27	370	29	368	35	362	37
393	6	399	4	385	14	391	12	377	22	383	20	369	30	375	28	361	38	367	36

				390	15	388	9
				387	10	389	16
				13	392	11	386
				12	385	14	391


47	356	41	358	55	348	49	350	63	340	57	342	71	332	65	334
42	357	48	355	50	349	56	347	58	341	64	339	66	333	72	331
360	43	354	45	352	51	346	53	344	59	338	61	336	67	330	69
353	46	359	44	345	54	351	52	337	62	343	60	329	70	335	68

								64	339	58	341
								338	61	344	59
								343	60	337	62
								57	342	63	340

By shifting the 4x4 magic square on a 2x2 carpet we take care that the numbers 1 and 63 respectively 12 and 20 will be put opposite to each other, so by swapping the digits the difference of +/- 62 in the rows and the difference of +/- 8 in the columns of the half borders can be corrected.

103	132	182	197	233	269	284	312	64	339	58	341	96	117	139	168	211	219	255	291
183	198	234	270	278	104	133	311	338	61	344	59	97	169	212	220	256	285	118	140
235	264	279	105	134	184	199	310	31	372	25	374	98	221	250	286	119	141	170	213
280	106	135	185	200	229	265	309	26	373	32	371	99	287	120	142	171	214	215	251
136	186	194	230	266	281	107	308	376	27	370	29	100	143	172	208	216	252	288	121
195	231	267	282	108	137	180	307	369	30	375	28	101	209	217	253	289	122	144	166
268	283	109	131	181	196	232	306	343	60	337	62	102	254	290	123	138	167	210	218
328	327	326	325	324	323	322	88	57	342	1	340	80	87	86	85	84	83	82	320
23	380	39	364	33	366	17	382	71	332	65	334	390	15	47	356	41	358	388	9
18	381	34	365	40	363	24	379	66	333	72	331	387	10	42	357	48	355	389	16
384	19	368	35	362	37	378	21	336	67	330	69	13	392	360	43	354	45	11	386
377	22	361	38	367	36	383	12	329	70	335	68	20	385	353	46	359	44	14	391
73	74	75	76	77	78	79	321	7	396	63	398	313	314	315	316	317	318	319	81
124	153	175	204	226	248	277	89	2	397	8	395	305	110	146	161	190	240	262	298
176	205	227	249	271	125	154	90	55	348	49	350	304	162	191	241	263	292	111	147
228	243	272	126	155	177	206	91	50	349	56	347	303	242	257	293	112	148	163	192
273	127	156	178	207	222	244	92	352	51	346	53	302	294	113	149	164	193	236	258
157	179	201	223	245	274	128	93	345	54	351	52	301	150	165	187	237	259	295	114
202	224	246	275	129	158	173	94	400	3	394	5	300	188	238	260	296	115	151	159
247	276	130	152	174	203	225	95	393	6	399	4	299	261	297	116	145	160	189	239

The 20x20 inlay is ready now.

The 22x22 border

To construct the 22x22 border we use the numbers 1 up to 42 and 443 up to 484 (translated into -/- 42 up to -/- 1). See the following steps:

+/+	22	23	1	42	2	41	3	40	4	39	20	237
-/-	22	5	38	6	37	7	36	8	35	9	34	237
+/+	23	10	33	11	32	12	31	13	30	14	28	237
-/-	15	16	17	18	19	21	24	25	26	27	29	237

22	1	42	2	41	3	40	4	39	20	-15	-16	-17	-18	-19	-21	-24	-25	-26	-27	-29	23
																					10
																					33
																					11
																					32
																					12
																					31
																					13
																					30
																					14
																					28
																					-5
																					-38
																					-6
																					-37
																					-7
																					-36
																					-8
																					-35
																					-9
																					-34
																					-22

22	1	42	2	41	3	40	4	39	20	-15	-16	-17	-18	-19	-21	-24	-25	-26	-27	-29	23
-10																					10
-33																					33
-11																					11
-32																					32
-12																					12
-31																					31
-13																					13
-30																					30
-14																					14
-28																					28
5																					-5
38																					-38
6																					-6
37																					-37
7																					-7
36																					-36
8																					-8
35																					-35
9																					-9
34																					-34
-23	-1	-42	-2	-41	-3	-40	-4	-39	-20	15	16	17	18	19	21	24	25	26	27	29	-22

2	1	42	2	41	3	40	4	39	20	470	469	468	467	466	464	461	460	459	458	456	23
475																					10
452																					33
474																					11
453																					32
473																					12
454																					31
472																					13
455																					30
471																					14
457																					28
5																					480
38																					447
6																					479
37																					448
7																					478
36																					449
8																					477
35																					450
9																					476
34																					451
462	484	443	483	444	482	445	481	446	465	15	16	17	18	19	21	24	25	26	27	29	463

The final result

Add 42 to all numbers of the 20x20 inlay and combine the 22x22 border with the 20x20 inlay:

22x22 inlaid square

22	1	42	2	41	3	40	4	39	20	470	469	468	467	466	464	461	460	459	458	456	23
475	145	174	224	239	275	311	326	354	106	381	100	383	138	159	181	210	253	261	297	333	10
452	225	240	276	312	320	146	175	353	380	103	386	101	139	211	254	262	298	327	160	182	33
474	277	306	321	147	176	226	241	352	73	414	67	416	140	263	292	328	161	183	212	255	11
453	322	148	177	227	242	271	307	351	68	415	74	413	141	329	162	184	213	256	257	293	32
473	178	228	236	272	308	323	149	350	418	69	412	71	142	185	214	250	258	294	330	163	12
454	237	273	309	324	150	179	222	349	411	72	417	70	143	251	259	295	331	164	186	208	31
472	310	325	151	173	223	238	274	348	385	102	379	104	144	296	332	165	180	209	252	260	13
455	370	369	368	367	366	365	364	130	99	384	43	382	122	129	128	127	126	125	124	362	30
471	65	422	81	406	75	408	59	424	113	374	107	376	432	57	89	398	83	400	430	51	14
457	60	423	76	407	82	405	66	421	108	375	114	373	429	52	84	399	90	397	431	58	28
5	426	61	410	77	404	79	420	63	378	109	372	111	55	434	402	85	396	87	53	428	480
38	419	64	403	80	409	78	425	54	371	112	377	110	62	427	395	88	401	86	56	433	447
6	115	116	117	118	119	120	121	363	49	438	105	440	355	356	357	358	359	360	361	123	479
37	166	195	217	246	268	290	319	131	44	439	50	437	347	152	188	203	232	282	304	340	448
7	218	247	269	291	313	167	196	132	97	390	91	392	346	204	233	283	305	334	153	189	478
36	270	285	314	168	197	219	248	133	92	391	98	389	345	284	299	335	154	190	205	234	449
8	315	169	198	220	249	264	286	134	394	93	388	95	344	336	155	191	206	235	278	300	477
35	199	221	243	265	287	316	170	135	387	96	393	94	343	192	207	229	279	301	337	156	450
9	244	266	288	317	171	200	215	136	442	45	436	47	342	230	280	302	338	157	193	201	476
34	289	318	172	194	216	245	267	137	435	48	441	46	341	303	339	158	187	202	231	281	451
462	484	443	483	444	482	445	481	446	465	15	16	17	18	19	21	24	25	26	27	29	463

22x22, Inlaid.xls

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22x22 magic square