---

950

The smallest squares containing k 950's :
559504 = 748²,
9504495081 = 97491²,
9509507895001 = 3083749².

950² + 951² + 952² + ... + 13248² = 880257².

(1² + 2² + ... + 300²)(301² + 302² + ... + 924²)(925² + 926² + ... + 950²) = 229292017500².

The 4-by-4 magic square consisting of different squares with constant 950:

Page of Squares : First Upload February 20, 2006 ; Last Revised October 30, 2009
by Yoshio Mimura, Kobe, Japan

---

951

The smallest squares containing k 951's :
1595169 = 1263²,
9519514624 = 97568²,
209514595195161 = 14474619².

Komachi cube sum : 951² = 5³ + 7³ + 8³ + 12³ + 46³ + 93³.

(1³ + 2³ + ... + 776³)(777³ + 778³ + ... + 951³) = 101802415680².

3-by-3 magic squares consisting of different squares with constant 951²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(2, 179, 934, 466, 814, 157, 829, 458, 86),	(2, 179, 934, 586, 746, 67, 749, 562, 166),
(2, 194, 931, 586, 733, 154, 749, 574, 118),	(9, 168, 936, 648, 684, 129, 696, 639, 108),
(14, 541, 782, 637, 586, 394, 706, 518, 371),	(19, 178, 934, 262, 899, 166, 914, 254, 67),
(19, 418, 854, 574, 686, 323, 758, 509, 266),	(26, 326, 893, 542, 739, 254, 781, 502, 206),
(26, 563, 766, 598, 586, 451, 739, 494, 338),	(29, 154, 938, 658, 674, 131, 686, 653, 86),
(33, 396, 864, 576, 696, 297, 756, 513, 264),	(35, 470, 826, 670, 574, 355, 674, 595, 310),
(46, 173, 934, 509, 794, 122, 802, 494, 131),	(67, 206, 926, 586, 739, 122, 746, 562, 179),
(74, 163, 934, 221, 914, 142, 922, 206, 109),	(74, 430, 845, 605, 674, 290, 730, 515, 326),
(86, 298, 899, 557, 746, 194, 766, 509, 242),	(98, 461, 826, 646, 574, 397, 691, 602, 254),
(109, 614, 718, 662, 466, 499, 674, 557, 374),	(144, 492, 801, 648, 639, 276, 681, 504, 432),
(154, 334, 877, 371, 838, 254, 862, 301, 266),	(156, 441, 828, 612, 684, 249, 711, 492, 396),
(158, 394, 851, 499, 766, 262, 794, 403, 334),	(166, 562, 749, 614, 509, 518, 707, 574, 274),
(206, 418, 829, 466, 781, 278, 803, 346, 374),	(250, 499, 770, 530, 730, 301, 749, 350, 470)

951² = 904401 appears in the decimal expression of π:
  π = 3.14159•••904401••• (from the 81690th digit).

Page of Squares : First Upload February 20, 2006 ; Last Revised October 30, 2009
by Yoshio Mimura, Kobe, Japan

---

952

The smallest squares containing k 952's :
79524 = 282²,
1829529529 = 42773²,
74589529529529 = 8636523².

(1² + 2² + ... + 104²)(105²)(106² + 107² + ... + 952²) = 1098347250².

(1³ + 2³ + ... + 832³)(833³ + 834² + ... + 952³) = 101442606720².

952² = 906304, 9 + 0 + 6 + 30 + 4 = 7²,
952² = 906304, 90 + 6 + 304 = 20².

952² = 906304 appears in the decimal expression of e:
  e = 2.71828•••906304••• (from the 56417th digit).

Page of Squares : First Upload February 20, 2006 ; Last Revised October 13, 2006
by Yoshio Mimura, Kobe, Japan

---

953

The smallest squares containing k 953's :
59536 = 244²,
495329536 = 22256²,
39953953795396 = 6320914².

953² = 431² + 432² + 732² : 237² + 234² + 134² = 359².

Komachi fraction : 953² = 65391048 / 72.

3-by-3 magic squares consisting of different squares with constant 953²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(12, 233, 924, 584, 732, 177, 753, 564, 152),	(12, 252, 919, 368, 849, 228, 879, 352, 108),
(17, 96, 948, 276, 908, 87, 912, 273, 44),	(17, 492, 816, 564, 663, 388, 768, 476, 303),
(24, 528, 793, 633, 584, 408, 712, 537, 336),	(28, 177, 936, 663, 676, 108, 684, 648, 143),
(28, 432, 849, 471, 732, 388, 828, 431, 192),	(72, 296, 903, 647, 648, 264, 696, 633, 152),
(72, 487, 816, 537, 696, 368, 784, 432, 327),	(80, 303, 900, 465, 800, 228, 828, 420, 215),
(87, 324, 892, 604, 672, 303, 732, 593, 144),	(145, 528, 780, 660, 620, 297, 672, 495, 460),
(152, 399, 852, 591, 712, 228, 732, 492, 361),	(156, 297, 892, 408, 836, 207, 847, 348, 264)

953² = 908209, 9 + 0 + 82 + 0 + 9 = 10².

Page of Squares : First Upload February 20, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

---

954

The smallest squares containing k 954's :
95481 = 309²,
954995409 = 30903²,
1954995495439561 = 44215331².

954² = (26 + 27 + 28)² + (29 + 30 + 31)² + (32 + 33 + 34)² + ... + (95 + 96 + 97)².

(541 / 954)² = 0.321586479... (Komachic).

Komachi square sum : 954² = 3² + 7² + 9² + 421² + 856².

The 4-by-4 magic squares consisting of different squares with constant 954:

Page of Squares : First Upload February 20, 2006 ; Last Revised October 30, 2009
by Yoshio Mimura, Kobe, Japan

---

955

The smallest squares containing k 955's :
695556 = 834²,
9559559529 = 97773²,
95559559558849 = 9775457².

955² is the fifth squares which is the sum of 10 seventh powers.

3-by-3 magic squares consisting of different squares with constant 955²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(26, 393, 870, 510, 730, 345, 807, 474, 190),	(30, 215, 930, 345, 870, 190, 890, 330, 105),
(30, 215, 930, 505, 786, 198, 810, 498, 89),	(30, 215, 930, 615, 726, 82, 730, 582, 201),
(30, 345, 890, 386, 810, 327, 873, 370, 114),	(30, 345, 890, 615, 674, 282, 730, 582, 201),
(30, 505, 810, 615, 630, 370, 730, 510, 345),	(54, 422, 855, 478, 729, 390, 825, 450, 170),
(55, 210, 930, 390, 855, 170, 870, 370, 135),	(55, 390, 870, 618, 649, 330, 726, 582, 215),
(82, 201, 930, 474, 818, 135, 825, 450, 170),	(98, 414, 855, 630, 615, 370, 711, 602, 210),
(105, 190, 930, 270, 903, 154, 910, 246, 153),	(105, 330, 890, 406, 825, 258, 858, 350, 231),
(105, 330, 890, 630, 694, 183, 710, 567, 294),	(105, 566, 762, 630, 615, 370, 710, 462, 441),
(135, 370, 870, 422, 810, 279, 846, 345, 278),	(135, 370, 870, 642, 681, 190, 694, 558, 345),
(162, 359, 870, 534, 762, 215, 775, 450, 330),	(162, 510, 791, 665, 630, 270, 666, 505, 462)

955² = 912025, and 9 = 3², 1 = 1², 2025 = 15².

955² = 912025, 91 + 2 + 0 + 2 + 5 = 10²,
955² = 912025, 91 + 2025 = 46².

Page of Squares : First Upload February 20, 2006 ; Last Revised October 30, 2009
by Yoshio Mimura, Kobe, Japan

---

956

The smallest squares containing k 956's :
17956 = 134²,
3379561956 = 58134²,
201956329565956 = 14211134².

The squares which begin with 956 and end in 956 are
956221913956 = 977866²,   956746121956 = 978134²,   9561292673956 = 3092134²,
9562727477956 = 3092366²,   9564385057956 = 3092634²,...

956² = 913936, a square with odd digits except the last digit 6.

Komachi equation : 956² = - 9⁶ + 8⁶ + 7⁶ + 6⁶ + 5⁶ + 4⁶ - 3⁶ - 2⁶ + 10⁶.

956² + 957² + 958² + ... + 1051² = 9836²,
956² + 957² + 958² + ... + 1677² = 35815²,
956² + 957² + 958² + ... + 3283² = 107282²,
956² + 957² + 958² + ... + 113901² = 22193889²,
956² + 957² + 958² + ... + 122205² = 24664675².

956² = 913936, 9 + 13 + 93 + 6 = 11²,
956² = 913936, 913 + 936 = 43².

Page of Squares : First Upload February 20, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

---

957

The smallest squares containing k 957's :
1495729 = 1223²,
108495795769 = 329387²,
6957195736957636 = 83409806².

957² = 112² + 413² + 856² : 658² + 314² + 211² = 759²,
957² = 116² + 493² + 812² : 218² + 394² + 611² = 759².

The square root of 957 is 30. 9 3 5 4 16 5 9 6 5 16 0 3 9 ...,
and 30² = 9² + 3² + 5² + 4² + 16² + 5² + 9² + 6² + 5² + 16² + 0² + 3² + 9².

3-by-3 magic squares consisting of different squares with constant 957²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(8, 157, 944, 277, 904, 148, 916, 272, 53),	(8, 508, 811, 608, 629, 388, 739, 512, 328),
(18, 378, 879, 447, 774, 342, 846, 417, 162),	(28, 304, 907, 427, 808, 284, 856, 413, 112),
(29, 232, 928, 472, 811, 188, 832, 452, 139),	(29, 232, 928, 512, 788, 181, 808, 491, 148),
(42, 234, 927, 522, 783, 174, 801, 498, 162),	(43, 260, 920, 580, 725, 232, 760, 568, 125),
(52, 211, 932, 659, 668, 188, 692, 652, 109),	(53, 332, 896, 448, 784, 317, 844, 437, 112),
(56, 227, 928, 268, 896, 203, 917, 248, 116),	(56, 637, 712, 668, 536, 427, 683, 472, 476),
(63, 486, 822, 606, 657, 342, 738, 498, 351),	(76, 437, 848, 587, 692, 304, 752, 496, 323),
(80, 395, 868, 640, 668, 245, 707, 560, 320),	(81, 258, 918, 522, 783, 174, 798, 486, 207),
(92, 424, 853, 664, 643, 248, 683, 568, 356),	(109, 388, 868, 532, 749, 268, 788, 452, 301),
(116, 493, 812, 532, 644, 467, 787, 508, 196),	(136, 412, 853, 512, 757, 284, 797, 416, 328),
(148, 524, 787, 557, 692, 356, 764, 403, 412),	(172, 533, 776, 587, 556, 512, 736, 568, 227),
(203, 464, 812, 556, 728, 277, 752, 413, 424),	(203, 464, 812, 592, 707, 256, 724, 448, 437)

957² = 915849, 9 + 1 + 5 + 8 + 4 + 9 = 6²,
957² = 915849, 9 + 1 + 58 + 4 + 9 = 9²,
957² = 915849, 9 + 15 + 8 + 49 = 9²,
957² = 915849, 915 + 849 = 42².

957² = 915849 appears in the decimal expression of e:
  e = 2.71828•••915849••• (from the 10170th digit),
  (915849 is the ninth 6-digit square in the expression of e.)

Page of Squares : First Upload February 20, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

---

958

The smallest squares containing k 958's :
29584 = 172²,
6958395889 = 83417²,
2495807958795844 = 49958062².

958² = 917764, 9 + 177 + 6 + 4 = 14²,
958² = 917764, 91 + 7 + 7 + 64 = 13²,
958² = 917764, 917 + 764 = 41².

958² = 917764 appears in the decimal expression of π:
  π = 3.14159•••917764••• (from the 65824th digit).

Page of Squares : First Upload February 20, 2006 ; Last Revised October 13, 2006
by Yoshio Mimura, Kobe, Japan

---

959

The smallest squares containing k 959's :
295936 = 544²,
32995995904 = 181648²,
279599599597824 = 16721232².

959² = 919681, a zigzag square.

959² = 114² + 454² + 837² = 738² + 454² + 411².

3-by-3 magic squares consisting of different squares with constant 959²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(5, 234, 930, 666, 670, 165, 690, 645, 166),	(6, 138, 949, 507, 806, 114, 814, 501, 78),
(6, 347, 894, 459, 786, 302, 842, 426, 171),	(18, 194, 939, 426, 843, 166, 859, 414, 102),
(22, 354, 891, 549, 726, 302, 786, 517, 186),	(27, 94, 954, 346, 891, 78, 894, 342, 59),
(27, 366, 886, 626, 678, 261, 726, 571, 258),	(50, 366, 885, 534, 725, 330, 795, 510, 166),
(54, 363, 886, 402, 814, 309, 869, 354, 198),	(54, 454, 843, 491, 738, 366, 822, 411, 274),
(58, 186, 939, 309, 894, 158, 906, 293, 114),	(58, 606, 741, 654, 517, 474, 699, 534, 382),
(69, 166, 942, 282, 906, 139, 914, 267, 114),	(69, 562, 774, 654, 594, 373, 698, 501, 426),
(84, 399, 868, 672, 644, 231, 679, 588, 336),	(94, 261, 918, 594, 738, 149, 747, 554, 234),
(114, 437, 846, 549, 666, 418, 778, 534, 171),	(114, 454, 837, 571, 642, 426, 762, 549, 194),
(117, 554, 774, 606, 558, 491, 734, 549, 282),	(139, 318, 894, 474, 806, 213, 822, 411, 274),
(150, 491, 810, 590, 690, 309, 741, 450, 410),	(198, 626, 699, 661, 414, 558, 666, 597, 346)

959² = 919681, and 9 = 3², 196 = 14², 81 = 9².

959² = 919681, 91 + 9 + 68 + 1 = 13²,
959² = 919681, 91 + 96 + 8 + 1 = 14²,
959² = 919681, 919 + 681 = 40².

Page of Squares : First Upload February 20, 2006 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

---