The first non-zero square number.
The smallest squares containing k 1's are :
1 = 12, 121 = 112, 11881 = 1092, 1110916 = 10542,
11431161 = 33812, 111112681 = 105412, 11112111396 = 1054142,
171511111321 = 4141392, 1113111511681 = 10550412.
The squares which begin with 1 and end in 1 are
121 = 112, 1521 = 392, 1681 = 412, 10201 = 1012, 11881 = 1092,...
Komachi equations:
12 = 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 293 equations,
12 = 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 311 equations,
12 = 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 197 equations,
12 = 12 + 22 + 32 + 42 - 52 - 62 + 72 + 82 - 92 = 12 + 22 - 32 * 42 * 52 / 62 - 72 + 82 + 92
= 122 * 32 / 42 + 562 / 72 / 82 - 92, and more 14 equations,
12 = 92 / 82 * 72 - 62 - 52 - 42 / 322 * 12 = 92 + 82 * 72 - 652 - 42 + 322 + 12
= 92 - 82 + 72 - 62 - 52 - 42 + 32 + 22 - 12, and more 12 equations,
12 = 982 / 72 + 62 + 52 + 42 * 32 - 22 * 102 = 92 + 82 + 72 * 62 * 52 - 42 * 32 - 2102
= 92 + 82 - 72 * 62 * 52 - 42 * 32 + 2102, and more 3 equations,
12 = 123 * 33 / 43 + 563 / 73 / 83 - 93 = 123 * 33 / 43 * 563 / 73 / 83 / 93
= 13 / 23 * 33 * 43 - 53 - 63 + 73 + 83 - 93, and more 6 equations,
12 = - 93 + 83 + 73 - 63 - 53 + 43 * 33 / 23 * 13 = - 93 + 83 + 73 - 63 * 53 / 43 / 33 * 23 * 13
= - 93 + 83 + 73 + 63 - 53 - 43 * 33 / 23 * 13, and more 3 equations,
12 = 12t * 3t / 4t + 56t / 7t / 8t - 9t = 12t * 3t / 4t * 56t / 7t / 8t / 9t
= 12t * 3t / 4t / 56t * 7t * 8t / 9t = - 12t * 3t / 4t + 56t / 7t / 8t + 9t for any t.
Page of Squares : First Upload August 17, 2003 ; Last Revised May 18, 2010
by Yoshio Mimura, Kobe, Japan