The smallest squares containing k 22's :
225 = 152, 2202256 = 14842, 2212291225 = 470352,
2222222266944 = 14907122, 122222200822225 = 110554152.
The first integer which is the sum of 7 squares in just two ways (see 20).
Every integer greater than 22 is the sum of 9 nonzero squares.
222 = 484, 4 + 8 + 4 = 42.
222 = 2! + 2! + 5! + 5! + 5! + 5!.
222 = 484 is a palindromic square.
222 = 23 + 23 + 53 + 73.
222 = 484, every digit of which is even. There are only 3-digits squares consisting of even digits.
22k + 24k + 58k + 65k are squares for k = 1,2,3 (132, 932, 7032).
22k + 73k + 130k + 136k are squares for k = 1,2,3 (192, 2032, 22612).
22k + 161k + 266k + 280k are squares for k = 1,2,3 (272, 4192, 67052).
Komachi Fractions : (3/22)2 = 1539/82764 = 7695/413820, (22/31)2 = 13068/25947, (22/89)2 = 4356/71289.
Komachi equations:
222 = 1 + 2 + 3 + 4 - 5 * 6 + 7 * 8 * 9 = 1 + 2 + 3 - 4 * 5 - 6 + 7 * 8 * 9
= 1 + 2 - 3 * 4 - 5 - 6 + 7 * 8 * 9, and more 51 equations,
222 = 9 * 8 * 7 + 6 - 5 * 4 - 3 - 2 - 1 = 9 * 8 * 7 + 6 - 5 * 4 - 3 * 2 * 1
= 9 * 8 * 7 - 6 - 5 - 4 - 3 - 2 * 1, and more 42 equations,
222 = 9 * 8 * 7 + 6 - 5 - 4 + 3 - 2 * 10 = 9 * 8 * 7 - 6 + 5 + 4 - 3 - 2 * 10
= 9 * 8 * 7 - 6 + 5 - 4 - 3 - 2 - 10, and more 32 equations,
222 = 12 * 232 - 42 - 52 - 62 + 72 + 82 - 92 = 12 + 232 * 42 + 52 - 62 - 72 - 892
= 12 + 22 * 32 * 42 - 52 - 62 - 72 - 82 + 92, and more 3 equations,
222 = - 92 + 82 - 72 * 62 - 52 + 432 + 212 = 92 - 82 + 72 * 62 - 542 / 32 * 22 - 12
= - 92 + 82 + 72 + 62 * 52 - 42 + 32 - 212, and more 2 equations,
222 = 92 - 82 + 72 + 62 - 52 + 42 - 32 + 22 * 102 = 92 - 82 * 72 - 62 - 52 + 42 * 32 / 22 * 102
222 = 982 - 762 * 52 / 42 + 32 - 22 - 102.
228 = 54875873536, 5 + 48 + 7 + 58 + 7 + 353 + 6 = 222,
2210 = 26559922791424, 22 + 62 + 52 + 592 + 92 + 22 + 22 + 792 + 142 + 242 = 223,
2211 = 584318301411328, 5 + 8 + 4 + 3 + 1 + 8 + 3 + 01 + 411 + 32 + 8 = 222 and more 43 equations,
(222 + 2) = (42 + 2)(52 + 2) = (12 + 2)(22 + 2)(52 + 2),
(222 - 4) = (32 - 4)(102 - 4), (222 + 6)= (12 + 6)(82 + 6),
(222 - 8) = (52 - 8)(62 - 8).
222 + 232 + 242 + ... + 69102 = 3316682,
222 + 232 + 242 + ... + 802 = 4132.
(12 + 22 + 32 + 42)(52 + 62 + ... + 122)(132)(142 + 152 + ... + 222) = 967202.
12 + 22 + 32 + ... + 222 = 3795, which consists of different odd digits (the second 4-digit sum).
13 + 23 + 33 + 43 + ... + 223 = (1 + 2 + 3 + 4 + ... + 22)2 = 2532.
222 = 484 appears in the decimal expressions of π and e:
π = 3.14159•••484••• (from the 2363rd digit),
e = 2.71828•••484••• (from the 781st digit).
Page of Squares : First Upload October 13, 2003 ; Last Revised January 25, 2011
by Yoshio Mimura, Kobe, Japan