1.+Squares+and+Cubes

= 1729 = = A curious number = Indeed: 1729 = 1 3 + 12 3  = 9 3 + 10 3 He really was good at doing sums in his head! || || We would like you to investigate this by expressing the numbers from 1 to 200 as sums of perfect squares. You can express as many as you like but you cannot do the task for consecutive numbers. || ||
 * [|**Srinivasa Ramanujan**](1887-1920), the Indian mathematician, went to England in 1913 to work with the English mathematician **[|G. H. Hardy]**. Unfortunately, he soon became ill. Hardy went to visit him one day, and when he arrived he said that he had travelled by taxi and that the number on the taxi, 1729, was very boring. Ramanujan answered, "No, it is a very interesting number! It is the smallest number that you can express as the sum of two cubes in two different ways."
 * The famous French mathematicain, [|**Lagrange**](1736-1813) proved that you could express any natural number as the sum of at most four perfect squares.

3² || 25+9 ||= Marc J ||= Easy ||
 * = ==Number - == ||= == -- Expression 1 -- == ||= ==Expression 2== ||= ==Initials== ||= ==Comments== ||
 * = 1 ||= 1 2 ||= 1 ||= James W. ||= Easy ||
 * = 2 ||= 1 2 +1 2 ||= 1+1 ||= James W. ||= Easy ||
 * = 3 ||= 1²+1²+1² ||= 1+1+1 ||= Anna M. ||= Easy ||
 * = 4 ||= 2² ||= 2+2 ||= Anna M. ||= Easy ||
 * = 5 ||= 2²+1² ||= 4+1 ||= Anna M. ||= Easy ||
 * = 6 ||= 2²+1²+1² ||= 4+1+1 ||= Sara E. ||= Easy ||
 * = 7 ||= 2²+1²+1²+1² ||= 4+1+1+1 ||= Sara E. ||= Easy ||
 * = 8 ||= 2²+2² ||= 4+4 ||= Sara E. ||= Easy ||
 * = 9 ||= 2²+2²+1² ||= 4+4+1 ||= Sara E. ||= Easy ||
 * = 10 ||= 1²+3² ||= 1+9 ||= Sara E. ||= Easy ||
 * = 11 ||= 1²+1²+3² ||= 1+1+9 ||= Helena Ll. ||= Easy ||
 * = 12 || 1²+1²+1²+3² || 1+1+1+9 ||= Helena Ll. ||= Easy ||
 * = 13 ||~ 3²+2² ||= 9+4 ||= Helena Ll. ||= Easy ||
 * = 14 || 3²+2²+1² || 9+4+1 ||= Helena Ll. ||= Easy ||
 * = 15 ||= 3²+2²+1²+1² ||= 9+4+1+1 ||= Helena Ll. ||= Easy ||
 * = 16 ||= 4² ||= 16 ||= Sara E. ||= Easy ||
 * = 17 ||= 4²+1² ||= 16+1 ||= Helena Ll. ||= Easy ||
 * = 18 ||= 4²+1²+1² ||= 16+1+1 ||= Helena Ll. ||= Easy ||
 * = 19 ||= 4²+1²+1²+1² ||= 16+1+1+1 ||= Helena Ll. ||= Easy ||
 * = 20 ||= 4²+2² ||= 16+4 ||= Helena Ll. ||= Easy ||
 * = 21 ||= 4²+2²+1² ||= 16+4+1 ||= Helena Ll. ||= Easy ||
 * = 22 || 4²+2²+1²+1² || 16+4+1+1 ||= Helena Ll. ||= Easy ||
 * = 23 || 4²+2²+1²+1²+1² || 16+4+1+1+1 ||= Helena Ll. ||= Easy ||
 * = 24 || 4²+2²+2² || 16+4+4 ||= Helena Ll. ||= Easy ||
 * = 25 ||= 5² ||= 25 ||= Helena Ll. ||= Easy ||
 * = 26 ||= 5²+1² ||= 25+1 ||= Helena Ll. ||= Easy ||
 * = 27 || 5²+1²+1² || 25+1+1 ||= Mar L. ||= Easy ||
 * = 28 || 5²+1²+1²+1² || 25+1+1+1 ||= Mar L. ||= Easy ||
 * = 29 ||= 5²+2² ||= 25+4 ||= Mar L. ||= Easy ||
 * = 30 || 52+22+12 || 25+4+1 ||= Marc J ||= Easy ||
 * = 31 ||= 52+22+12+12 ||= 25+4+1+1 ||= Marc J ||= Easy ||
 * = 32 || 42+42 || 16+16 ||= Marc J ||= Easy ||
 * = 33 || 42+42+12 || 16+16+1 ||= Marc J ||= Easy ||
 * = 34 || 5² +
 * = 35 || 52+32+12 || 25+9+1 ||= Marc J ||= Easy ||
 * = 36 ||= 32x22 ||= 9x4 ||= Marc J ||= Easy ||
 * = 37 ||= 6²+1² ||= 36+1 ||= Mar L. ||= Easy ||
 * = 38 || 6²+1²+1² || 36+1+1 ||= Mar L. ||= Easy ||
 * = 39 || 6²+1²+1²+1² || 36+1+1+1 ||= Mar L. ||= Easy ||
 * = 40 || 6²+2² || 36+4 ||= Mar L. ||= Easy ||
 * = 41 ||= 6²+2²+1² ||= 36+4+1 ||= Mar L. ||= Easy ||
 * = 42 || 6²+2²+1²+1² || 36+4+1+1 ||= Mar L. ||= Easy ||
 * = 43 || 6²+2²+1²+1²+1² || 36+4+1+1+1 ||= Mar L. ||= Easy ||
 * = 44 || 6²+2²+2² || 36+4+4 ||= Mar L. ||= Easy ||
 * = 45 ||= 6²+3² ||= 36+9 ||= Mar L. ||= Easy ||
 * = 46 || 6²+3²+1² || 36+9+1 ||= Mar L. ||= Easy ||
 * = 47 ||= 6²+3²+1²+1² ||= 36+9+1+1 ||= Mar L. ||= Easy ||
 * = 48 || 6²+3²+1²+1²+1² || 36+9+1+1+1 ||= Mar L. ||= Easy ||
 * = 49 ||= 7² ||= 49 ||= Mar L. ||= Easy ||
 * = 50 ||= 7²+1² ||= 49+1 ||= Helena Ll. ||= Easy ||
 * = 51 || 7²+1²+1² || 49+1+1 ||= Helena Ll. ||= Easy ||
 * = 52 ||= 7²+1²+1²+1² ||= 49+1+1+1 ||= JaumeM. ||= Easy ||
 * = 53 ||= 7²+2² ||= 49+4 ||= JaumeM. ||= Easy ||
 * = 54 ||= 7²+2²+1² ||= 49+4+1 ||= JaumeM. ||= Easy ||
 * = 55 || 7²+2²+1²+1² || 49+4+1+1 ||= JaumeM. ||= Easy ||
 * = 56 || 7²+2²+1²+1²+1² || 49+4+1+1+1 ||= JaumeM. ||= easy ||
 * = 57 || 7²+2²+2² || 49+4+4 ||= jaumem. ||= easy ||
 * = 58 ||= 7²+3² ||= 49+9 ||= jaume.m ||= easy ||
 * = 59 ||= 7²+3²+1² ||= 49+9+1 ||= jaumem. ||= easy ||
 * = 60 ||= 7²+3²+1²+1² ||= 49+9+1+1 ||= jaumem. ||= easy ||
 * = 61 ||= 7²+3²+1²+1²+1² ||= 49+9+1+1+1 ||= jaumem. ||= easy ||
 * = 62 || 7²+3²+2² || 49+9+4 ||= jaumem. ||= Easy ||
 * = 63 || 7²+3²+2²+1² || 49+9+4+1 ||= jaumem. ||= Easy ||
 * = 64 ||= 8² ||= 64 ||= Mar L. ||= Easy ||
 * = 65 ||= 8²+1² ||= 64+1 ||= jaumem. ||= Easy ||
 * = 66 ||= 8²+1²+1² ||= 64+1+1 ||= jaumem. ||= Easy ||
 * = 67 ||= 8²+1²+1²+1² ||= 64+1+1+1 ||= jaumem. ||= Easy ||
 * = 68 ||= 8²+2² ||= 64+4 ||= jaumem. ||= Easy ||
 * = 69 ||= 8²+2²+1² ||= 64+4+1 ||= jaumem. ||= Easy ||
 * = 70 ||= 8²+2²+1²+1² ||= 64+4+1+1 ||= jaumem. ||= Easy ||
 * = 71 ||= 8²+2²+1²+1²+1² ||= 64+4+1+1+1 ||= jaumem. ||= Easy ||
 * = 72 ||= 8²+2²+2² ||= 64+4+4 ||= jaumem. ||= Easy ||
 * = 73 ||= 8²+3² ||= 64+9 ||= jaumem. ||= Easy ||
 * = 74 ||= 8²+3²+1²+1² ||= 64+9+1 ||= jaumeme. ||= Easy ||
 * = 75 ||= 8²+3²+1²+1² ||= 64+9+1+1 ||= jaumem. ||= Easy ||
 * = 76 ||= 8²+3²+1²+1²+1² ||= 64+9+1+1+1 ||= jaumem. ||= Easy ||
 * = 77 ||= 8²+3²+2² ||= 64+9+4 ||= jaumem. ||= Easy ||
 * = 78 ||= 8²+3²+2²+1² ||= 64+9+4+1 ||= jaumem. ||= Easy ||
 * = 79 ||= 8²+3²+2²+1²+1² ||= 64+9+4+1+1 ||= jaumem. ||= Easy ||
 * = 80 || 8²+3²+2²+1²+1²+1² || 64+9+4+1+1+1 ||= jaumem. ||= Easy ||
 * = 81 ||= 9² || 81 ||= Mar L. ||= Easy ||
 * = 82 ||= 9²+1² ||= 81+1 ||= jaumem. ||= Easy ||
 * = 83 ||= 9²+1²+1² ||= 81+1+1 ||= jaumem. ||= Easy ||
 * = 84 ||= 9²+1²+1²+1² ||= 81+1+1+1 ||= jaumem. ||= Easy ||
 * = 85 ||= 9²+2² ||= 81+4 ||= jaumem. ||= Easy ||
 * = 86 ||= 9²+2²+1² ||= 81+4+1 ||= jaumem. ||= Easy ||
 * = 87 ||= 9²+2²+1²+1² ||= 81+4+1+1 ||= jaumem. ||= Easy ||
 * = 88 ||= 9²+2²+1²+1²+1² ||= 81+4+1+1+1 ||= jaumem. ||= Easy ||
 * = 89 ||= 9²+2²+2² ||= 81+4+4 ||= jaumem. ||= Easy ||
 * = 90 ||= 9²+3² ||= 81+9 ||= jaumem. ||= Easy ||
 * = 91 ||= 9²+3²+1² ||= 81+9+1 ||= jaumem. ||= Easy ||
 * = 92 ||= 9²+3²+1²+1² ||= 81+9+1+1 ||= jaumem. ||= Easy ||
 * = 93 ||= 9²+3²+1²+1²+1² ||= 81+9+1+1+1 ||= jaumem. ||= Easy ||
 * = 94 ||= 9²+3²+2² ||= 81+9+4 ||= jaumem. ||= Easy ||
 * = 95 ||= 9²+3²+2²+1² ||= 81+9+4+1 ||= jaumem. ||= Easy ||
 * = 95 ||= 9²+3²+2²+1² ||= 81+9+4+1 ||= jaumem. ||= Easy ||
 * = 96 ||= 9²+3²+2²+1²+1² ||= 81+9+4+1+1 ||= jaumem. ||= Easy ||
 * = 97 ||= 9²+3²+2²+1²+1²+1² ||= 81+9+4+1+1+1 ||= jaumem. ||= Easy ||
 * = 98 ||= 9²+3²+2²+2² ||= 81+9+4+4 ||= jaumem. ||= Easy ||
 * = 99 || 9²+3²+3² || 81+9+9 ||= Helena Ll. ||= Easy ||
 * = 100 ||= 10² || 100 ||= Helena Ll. ||= Easy ||
 * = 101 ||= 10²+1² || 100+1 ||= Helena Ll. ||= Easy ||
 * = 102 || 10²+1²+1² || 100+1+1 ||= Helena Ll. ||= Easy ||
 * = 103 || 10²+1²+1²+1² || 100+1+1+1 ||= Helena Ll. ||= Easy ||
 * = 104 ||= 10²+2² ||= 100+4 ||= Helena Ll. ||= Easy ||
 * = 105 ||= 10²+2²+1² ||= 100+4+1 ||= Helena Ll. ||= Easy ||
 * = 106 ||= 10²+2²+1²+1² ||= 100+4+1+1 ||= Helena Ll. ||= Easy ||
 * = 107 || 9²+3²+1²+4² || 81+9+1+16 ||= Renata.C ||= Easy ||
 * = 108 ||= 102+22+22 ||= 100+4+4 ||= victor h ||= Easy ||
 * = 109 ||= 102+22+22+12 ||= 100+4+4+1 ||= victorh ||= easy ||
 * = 110 ||= 102+22+22+12+12 ||= 100+4+4+1+1 ||= victor h ||= easy ||
 * = 111 ||=  ||=   ||=   ||=   ||
 * = 112 ||=  ||=   ||=   ||=   ||
 * = 113 ||=  ||=   ||=   ||=   ||
 * = 114 ||=  ||=   ||=   ||=   ||
 * = 115 ||=  ||=   ||=   ||=   ||
 * = 116 ||  ||   ||=   ||=   ||
 * = 117 ||=  ||=   ||=   ||=   ||
 * = 118 ||=  ||=   ||=   ||=   ||
 * = 119 ||=  ||=   ||=   ||=   ||
 * = 120 ||= 10²+4²+1²+1²+1²+1² ||= 100+16+1+1+1+1 ||= Pol P.M ||= Easy ||
 * = 121 ||=  ||=   ||=   ||=   ||
 * = 122 ||=  ||=   ||=   ||=   ||
 * = 123 ||=  ||=   ||=   ||=   ||
 * = 124 ||  ||   ||=   ||=   ||
 * = 125 ||< 15² ||= 125 ||= Helena Ll. ||= Easy ||
 * = 126 ||= 15²+1² ||= 125+1 ||= Helena Ll. ||= Easy ||
 * = 127 ||= 15²+1²+1² ||= 125+1+1 ||= Helena Ll. ||= Easy ||
 * = 128 ||  ||   ||=   ||=   ||
 * = 129 ||=  ||=   ||=   ||=   ||
 * = 130 ||=  ||=   ||=   ||=   ||
 * = 131 ||=  ||=   ||=   ||=   ||
 * = 132 ||=  ||=   ||=   ||=   ||
 * = 133 ||  ||   ||=   ||=   ||
 * = 134 ||=  ||=   ||=   ||=   ||
 * = 135 ||=  ||=   ||=   ||=   ||
 * = 136 ||=  ||=   ||=   ||=   ||
 * = 137 ||=  ||=   ||=   ||=   ||
 * = 138 ||=  ||=   ||=   ||=   ||
 * = 140 ||  ||   ||=   ||=   ||
 * = 141 ||  ||   ||=   ||=   ||
 * = 142 ||=  ||=   ||=   ||=   ||
 * = 143 ||=  ||=   ||=   ||=   ||
 * = 144 ||=  ||=   ||=   ||=   ||
 * = 145 ||=  ||=   ||=   ||=   ||
 * = 146 ||=  ||=   ||=   ||=   ||
 * = 147 ||=  ||=   ||=   ||=   ||
 * = 148 ||=  ||=   ||=   ||=   ||
 * = 149 ||=  ||=   ||=   ||=   ||
 * = 150 ||  ||   ||=   ||=   ||
 * = 151 ||=  ||=   ||=   ||=   ||
 * = 152 ||=  ||=   ||=   ||=   ||
 * = 153 ||=  ||=   ||=   ||=   ||
 * = 154 ||=  ||=   ||=   ||=   ||
 * = 155 ||=  ||=   ||=   ||=   ||
 * = 156 ||=  ||=   ||=   ||=   ||
 * = 157 ||=  ||=   ||=   ||=   ||
 * = 158 ||=  ||=   ||=   ||=   ||
 * = 159 ||=  ||=   ||=   ||=   ||
 * = 160 ||=  ||=   ||=   ||=   ||
 * = 161 ||=  ||=   ||=   ||=   ||
 * = 162 ||=  ||=   ||=   ||=   ||
 * = 163 ||=  ||=   ||=   ||=   ||
 * = 164 ||=  ||=   ||=   ||=   ||
 * = 165 ||=  ||=   ||=   ||=   ||
 * = 166 ||=  ||=   ||=   ||=   ||
 * = 167 ||=  ||=   ||=   ||=   ||
 * = 168 ||  ||   ||=   ||=   ||
 * = 169 ||=  ||=   ||=   ||=   ||
 * = 170 ||=  ||=   ||=   ||=   ||
 * = 171 ||=  ||=   ||=   ||=   ||
 * = 172 ||=  ||=   ||=   ||=   ||
 * = 173 ||=  ||=   ||=   ||=   ||
 * = 174 ||=  ||=   ||=   ||=   ||
 * = 175 ||=  ||=   ||=   ||=   ||
 * = 176 ||=  ||=   ||=   ||=   ||
 * = 177 ||=  ||=   ||=   ||=   ||
 * = 178 ||=  ||=   ||=   ||=   ||
 * = 179 ||=  ||=   ||=   ||=   ||
 * = 180 ||=  ||=   ||=   ||=   ||
 * = 181 ||=  ||=   ||=   ||=   ||
 * = 182 ||=  ||=   ||=   ||=   ||
 * = 183 ||=  ||=   ||=   ||=   ||
 * = 184 ||=  ||=   ||=   ||=   ||
 * = 185 ||=  ||=   ||=   ||=   ||
 * = 186 ||=  ||=   ||=   ||=   ||
 * = 187 ||=  ||=   ||=   ||=   ||
 * = 188 ||=  ||=   ||=   ||=   ||
 * = 189 ||=  ||=   ||=   ||=   ||
 * = 190 ||=  ||=   ||=   ||=   ||
 * = 191 ||=  ||=   ||=   ||=   ||
 * = 192 ||=  ||=   ||=   ||=   ||
 * = 193 ||=  ||=   ||=   ||=   ||
 * = 194 ||=  ||=   ||=   ||=   ||
 * = 195 ||=  ||=   ||=   ||=   ||
 * = 196 ||= 14² ||= 196 ||= AdaG. ||= Easy ||
 * = 197 ||=  ||=   ||=   ||=   ||
 * = 198 ||=  ||=   ||=   ||=   ||
 * = 199 || 10²+9²+3²+3² || 100+81+9+9 ||= Helena Ll. ||= Easy ||
 * = 200 ||= 10²+10² ||= 100+100 ||= Helena Ll. ||= Easy ||