A 15x15 Magic Square

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 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 15x15 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (1695)

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 15 by 15 magic square shown below has a magic sum equal to 1695.
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 15-by-15 Magic Square

(Magic Sum = 1695)

Books about
Magic Squares
at Amazon.co.UK

Support Our Site
by purchasing our
Amazing Math Whiz
products
and other
Amazing Messages

122	139	156	173	190	207	224	1	18	35	52	69	86	103	120
138	155	172	189	206	223	15	17	34	51	68	85	102	119	121
154	171	188	205	222	14	16	33	50	67	84	101	118	135	137
170	187	204	221	13	30	32	49	66	83	100	117	134	136	153
186	203	220	12	29	31	48	65	82	99	116	133	150	152	169
202	219	11	28	45	47	64	81	98	115	132	149	151	168	185
218	10	27	44	46	63	80	97	114	131	148	165	167	184	201
9	26	43	60	62	79	96	113	130	147	164	166	183	200	217
25	42	59	61	78	95	112	129	146	163	180	182	199	216	8
41	58	75	77	94	111	128	145	162	179	181	198	215	7	24
57	74	76	93	110	127	144	161	178	195	197	214	6	23	40
73	90	92	109	126	143	160	177	194	196	213	5	22	39	56
89	91	108	125	142	159	176	193	210	212	4	21	38	55	72
105	107	124	141	158	175	192	209	211	3	20	37	54	71	88
106	123	140	157	174	191	208	225	2	19	36	53	70	87	104

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

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 (8) and (9) 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 15 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 (120) 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.

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.

We also have other
Amazing Message
stores.

Click each image to see
many amazing choices

Amazing Teacher

Amazing Student

Amazing Reader

Amazing Artist

122	139	156	173	190	207	224	1	18	35	52	69	86	103	120
138	155	172	189	206	223	15	17	34	51	68	85	102	119	121
154	171	188	205	222	14	16	33	50	67	84	101	118	135	137
170	187	204	221	13	30	32	49	66	83	100	117	134	136	153
186	203	220	12	29	31	48	65	82	99	116	133	150	152	169
202	219	11	28	45	47	64	81	98	115	132	149	151	168	185
218	10	27	44	46	63	80	97	114	131	148	165	167	184	201
9	26	43	60	62	79	96	113	130	147	164	166	183	200	217
25	42	59	61	78	95	112	129	146	163	180	182	199	216	8
41	58	75	77	94	111	128	145	162	179	181	198	215	7	24
57	74	76	93	110	127	144	161	178	195	197	214	6	23	40
73	90	92	109	126	143	160	177	194	196	213	5	22	39	56
89	91	108	125	142	159	176	193	210	212	4	21	38	55	72
105	107	124	141	158	175	192	209	211	3	20	37	54	71	88
106	123	140	157	174	191	208	225	2	19	36	53	70	87	104

122	139	156	173	190	207	224	1	18	35	52	69	86	103	120
138	155	172	189	206	223	15	17	34	51	68	85	102	119	121
154	171	188	205	222	14	16	33	50	67	84	101	118	135	137
170	187	204	221	13	30	32	49	66	83	100	117	134	136	153
186	203	220	12	29	31	48	65	82	99	116	133	150	152	169
202	219	11	28	45	47	64	81	98	115	132	149	151	168	185
218	10	27	44	46	63	80	97	114	131	148	165	167	184	201
9	26	43	60	62	79	96	113	130	147	164	166	183	200	217
25	42	59	61	78	95	112	129	146	163	180	182	199	216	8
41	58	75	77	94	111	128	145	162	179	181	198	215	7	24
57	74	76	93	110	127	144	161	178	195	197	214	6	23	40
73	90	92	109	126	143	160	177	194	196	213	5	22	39	56
89	91	108	125	142	159	176	193	210	212	4	21	38	55	72
105	107	124	141	158	175	192	209	211	3	20	37	54	71	88
106	123	140	157	174	191	208	225	2	19	36	53	70	87	104

A 15x15 Magic Square

Scroll down to see a magic squarein which all rows, columns, and both diagonalsadd up to the same magic sum (1695)

Read more about Magic Squares

A 15-by-15 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 (1695)

122	139	156	173	190	207	224	1	18	35	52	69	86	103	120
138	155	172	189	206	223	15	17	34	51	68	85	102	119	121
154	171	188	205	222	14	16	33	50	67	84	101	118	135	137
170	187	204	221	13	30	32	49	66	83	100	117	134	136	153
186	203	220	12	29	31	48	65	82	99	116	133	150	152	169
202	219	11	28	45	47	64	81	98	115	132	149	151	168	185
218	10	27	44	46	63	80	97	114	131	148	165	167	184	201
9	26	43	60	62	79	96	113	130	147	164	166	183	200	217
25	42	59	61	78	95	112	129	146	163	180	182	199	216	8
41	58	75	77	94	111	128	145	162	179	181	198	215	7	24
57	74	76	93	110	127	144	161	178	195	197	214	6	23	40
73	90	92	109	126	143	160	177	194	196	213	5	22	39	56
89	91	108	125	142	159	176	193	210	212	4	21	38	55	72
105	107	124	141	158	175	192	209	211	3	20	37	54	71	88
106	123	140	157	174	191	208	225	2	19	36	53	70	87	104