610
The smallest squares containing k 610's :
36100 = 1902,
161036100 = 126902,
206106106108816 = 143563962.
226k + 610k + 822k + 1478k are squares for k = 1,2,3 (562, 18122, 634242).
230k + 610k + 665k + 2720k are squares for k = 1,2,3 (652, 28752, 1437252).
(12 + 22 + ... + 4392) + (12 + 22 + ... + 5222) = (12 + 22 + ... + 6102).
Page of Squares : First Upload January 25, 2006 ; Last Revised March 17, 2011by Yoshio Mimura, Kobe, Japan
611
The smallest squares containing k 611's :
206116 = 4542,
51611661124 = 2271822,
2576116110746116 = 507554542.
(12 + 3)(22 + 3)(82 + 3)(142 + 3) = 6112 + 3.
(13 + 23 + ... + 1883)(1893 + 1903 + ... + 6113) = 33066079202.
6112 = 720 + 721 + 723.
3-by-3 magic squares consisting of different squares with constant 6112:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(1, 78, 606, 354, 494, 63, 498, 351, 46), | (1, 186, 582, 426, 417, 134, 438, 406, 129), |
(6, 143, 594, 242, 546, 129, 561, 234, 62), | (15, 330, 514, 370, 414, 255, 486, 305, 210), |
(18, 154, 591, 246, 543, 134, 559, 234, 78), | (26, 273, 546, 399, 406, 222, 462, 366, 161), |
(66, 399, 458, 426, 298, 321, 433, 354, 246), | (78, 234, 559, 321, 494, 162, 514, 273, 186), |
(78, 351, 494, 426, 386, 207, 431, 318, 294), | (81, 262, 546, 414, 426, 143, 442, 351, 234) |
6112 = 373321, 32 + 72 + 32 + 32 + 22 + 12 = 92,
6112 = 373321, 3 + 7 + 33 + 21 = 82,
6112 = 373321, 37 + 3 + 3 + 21 = 82,
6112 = 373321, 3 + 73 + 3 + 21 = 102,
6112 = 373321, 37 + 3 + 321 = 192.
6112 = 373321 appears in the decimal expression of π:
π = 3.14159•••373321••• (from the 62819th digit).
by Yoshio Mimura, Kobe, Japan
612
The smallest squares containing k 612's :
16129 = 1272,
612612001 = 247512,
61232612566129 = 78251272.
6122 = (12 + 8)(22 + 8)(32 + 8)(142 + 8) = (32 + 8)(102 + 8)(142 + 8).
Komachi equations
6122 = 12 * 22 * 342 * 562 / 72 / 82 * 92 = 12 * 22 * 342 / 562 * 72 * 82 * 92.
6122 = 64 + 124 + 124 + 244 = 44 + 84 + 144 + 244.
6122 + 6132 + 6142 + 6152 + ... + 6442 = 36082,
6122 + 6132 + 6142 + 6152 + ... + 19092 = 473772,
6122 + 6132 + 6142 + 6152 + ... + 22042 = 591182,
6122 + 6132 + 6142 + 6152 + ... + 35022 = 1193572,
6122 + 6132 + 6142 + 6152 + ... + 37902 = 1344532,
6122 + 6132 + 6142 + 6152 + ... + 46672 = 1838982,
6122 + 6132 + 6142 + 6152 + ... + 75252 = 3768132,
6122 + 6132 + 6142 + 6152 + ... + 95892 = 5420972,
6122 + 6132 + 6142 + 6152 + ... + 124112 = 7982702,
6122 + 6132 + 6142 + 6152 + ... + 1915802 = 484134822.
(1 + 2 + ... + 8)(9 + 10 + ... + 144) = 6122.
(13 + 23 + ... + 583)(593 + 603 + ... + 1173)(1183 + 1193 + ... + 6123) = 21449017636202.
Page of Squares : First Upload June 20, 2005 ; Last Revised December 7, 2013by Yoshio Mimura, Kobe, Japan
613
The smallest squares containing k 613's :
613089 = 7832,
344613613444 = 5870382,
61306613661316 = 78298542.
(415 / 613)2 = 0.458326791... (Komachic).
6132 = 375769, a zigzag square.
6133 - 6123 + 6113 - 6103 + ... + 13 = 107452.
6132 = 13 + 323 + 703, the 8th square which is the sum of three cubes.
6132 = 703 + 85 + 17.
3-by-3 magic squares consisting of different squares with constant 6132:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(12, 260, 555, 405, 420, 188, 460, 363, 180), | (27, 208, 576, 336, 477, 188, 512, 324, 93), |
(28, 216, 573, 384, 453, 152, 477, 352, 156), | (64, 372, 483, 408, 387, 244, 453, 296, 288), |
(72, 188, 579, 219, 552, 152, 568, 189, 132), | (72, 197, 576, 323, 504, 132, 516, 288, 163), |
(120, 315, 512, 387, 440, 180, 460, 288, 285) |
6132 = 375769, 37 + 5 + 7 + 6 + 9 = 82,
6132 = 375769, 3 + 7 + 5 + 76 + 9 = 102,
6132 = 375769, 3 + 75 + 7 + 6 + 9 = 102,
6132 = 375769, 3 + 7 + 5 + 769 = 282.
6132 = 375769 appears in the decimal expression of π:
π = 3.14159•••375769••• (from the 61862nd digit).
by Yoshio Mimura, Kobe, Japan
614
The smallest squares containing k 614's :
614656 = 7842,
361456144 = 190122,
614661461462689 = 247923672.
6142 = 376996, 3 + 7 + 6 + 9 + 96 = 112,
6142 = 376996, 3 + 7 + 6 + 99 + 6 = 112,
6142 = 376996, 37 + 69 + 9 + 6 = 112,
6142 = 376996, 376 + 9 + 9 + 6 = 202.
by Yoshio Mimura, Kobe, Japan
615
The smallest squares containing k 615's :
61504 = 2482,
35615615841 = 1887212,
1096156615361536 = 331082562.
Komachi square sum : 6152 = 32 + 962 + 1742 + 5822.
(13 + 23 + ... + 603)(613 + 623 + ... + 1643)(1653 + 1663 + ... + 6153) = 46350532800002.
6152 = 43 + 173 + 723.
3-by-3 magic squares consisting of different squares with constant 6152:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(0, 135, 600, 369, 480, 108, 492, 360, 81), | (10, 110, 605, 275, 542, 94, 550, 269, 58), |
(10, 110, 605, 418, 445, 74, 451, 410, 82), | (14, 77, 610, 173, 586, 70, 590, 170, 35), |
(14, 125, 602, 355, 490, 110, 502, 350, 61), | (19, 290, 542, 410, 445, 110, 458, 310, 269), |
(35, 170, 590, 310, 515, 130, 530, 290, 115), | (35, 218, 574, 310, 490, 205, 530, 301, 82), |
(35, 310, 530, 370, 413, 266, 490, 334, 163), | (35, 370, 490, 422, 371, 250, 446, 322, 275), |
(38, 290, 541, 334, 445, 262, 515, 310, 130), | (60, 207, 576, 360, 480, 135, 495, 324, 168), |
(70, 338, 509, 365, 434, 238, 490, 275, 250), | (74, 157, 590, 250, 550, 115, 557, 226, 130), |
(82, 205, 574, 350, 490, 125, 499, 310, 182), | (82, 326, 515, 410, 355, 290, 451, 382, 170), |
(110, 269, 542, 355, 458, 206, 490, 310, 205), | (115, 250, 550, 290, 514, 173, 530, 227, 214), |
(115, 290, 530, 394, 445, 158, 458, 310, 269) |
6152 = 378225, 37 + 82 + 25 = 122.
Page of Squares : First Upload June 20, 2005 ; Last Revised June 29, 2009by Yoshio Mimura, Kobe, Japan
616
The smallest squares containing k 616's :
41616 = 2042,
1616361616 = 402042,
291616961625616 = 170767962.
The squares which begin with 616 and end in 616 are
61605225616 = 2482042, 61650903616 = 2482962, 616545321616 = 7852042,
616689807616 = 7852962, 6161336697616 = 24822042,...
6162 = 379456, a square with different digits.
6162 = 379456, 3 + 7 + 945 + 6 = 312.
Page of Squares : First Upload June 20, 2005 ; Last Revised August 21, 2006by Yoshio Mimura, Kobe, Japan
617
The smallest squares containing k 617's :
76176 = 2762,
61717961761 = 2484312,
1061761761718596 = 325846862.
(13 + 23 + ... + 1363)(1373 + 1383 + ... + 4113)(4123 + 4133 + ... + 6173) = 1339176566019602.
6172 = 380689, 3 + 80 + 6 * 89 = 617.
3-by-3 magic squares consisting of different squares with constant 6172:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(12, 231, 572, 319, 492, 192, 528, 292, 129), | (24, 108, 607, 292, 537, 84, 543, 284, 72), |
(33, 140, 600, 360, 492, 95, 500, 345, 108), | (33, 184, 588, 248, 543, 156, 564, 228, 103), |
(39, 228, 572, 348, 481, 168, 508, 312, 159), | (60, 345, 508, 383, 420, 240, 480, 292, 255), |
(63, 212, 576, 432, 396, 193, 436, 423, 108), | (68, 336, 513, 423, 348, 284, 444, 383, 192), |
(72, 311, 528, 392, 432, 201, 471, 312, 248) |
6172 = 380689 appears in the decimal expression of π:
π = 3.14159•••380689••• (from the 82289th digit).
by Yoshio Mimura, Kobe, Japan
618
The smallest squares containing k 618's :
1361889 = 11672,
2618061889 = 511672,
618697618618689 = 248736332.
6182 = 381924, a zigzag square with different digits.
47586k + 79722k + 84666k + 169950k are squares for k = 1,2,3 (6182, 2113562, 782944202).
Komachi Square Sum : 6182 = 32 + 42 + 72 + 92 + 852 + 6122.
6182 = 332 + 523 + 593.
The 4-by-4 magic square consisting of different squares with constant 618:
|
6182 = 381924, 3 + 8 + 19 + 2 + 4 = 62,
6182 = 381924, 38 + 19 + 24 = 92.
by Yoshio Mimura, Kobe, Japan
619
The smallest squares containing k 619's :
236196 = 4862,
16198161984 = 1272722,
27636196196196 = 52570142.
240k + 306k + 313k + 366k are squares for k = 1,2,3 (352, 6192, 110532).
3-by-3 magic squares consisting of different squares with constant 6192:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(6, 282, 551, 313, 474, 246, 534, 281, 138), | (7, 126, 606, 174, 582, 119, 594, 169, 42), |
(9, 78, 614, 398, 471, 54, 474, 394, 57), | (18, 231, 574, 434, 414, 153, 441, 398, 174), |
(34, 246, 567, 378, 441, 214, 489, 358, 126), | (57, 286, 546, 394, 471, 78, 474, 282, 281), |
(114, 286, 537, 362, 471, 174, 489, 282, 254), | (142, 279, 534, 306, 506, 183, 519, 222, 254) |
6192 = 383161, 33 + 83 + 33 + 13 + 63 + 13 = 282,
6192 = 383161, 38 + 3 + 1 + 6 + 1 = 72,
6192 = 383161, 3 + 8 + 31 + 6 + 1 = 72,
6192 = 383161, 383 + 16 + 1 = 202,
6192 = 383161, 3 + 831 + 6 + 1 = 292.
by Yoshio Mimura, Kobe, Japan