More than one representation is often possible. Only one is given.
| 1 = 1 | 13 = 9 + 4 | 25 = 25 |
| 2 = 1 + 1 | 14 = 9 + 4 + 1 | 26 = 25 + 1 |
| 3 = 1 + 1 + 1 | 15 = 9 + 4 + 1 + 1 | 27 = 25 + 1 + 1 |
| 4 = 4 | 16 = 16 | 28 = 25 + 1 + 1 + 1 |
| 5 = 4 + 1 | 17 = 16 + 1 | 29 = 25 + 4 |
| 6 = 4 + 1 + 1 | 18 = 16 + 1 + 1 | 30 = 25 + 4 + 1 |
| 7 = 4 + 1 + 1 + 1 | 19 = 16 + 1 + 1 + 1 | 48 = 36 + 4 + 4 + 4 |
| 8 = 4 + 4 | 20 = 16 + 4 | 56 = 36 + 16 + 4 |
| 9 = 9 | 21 = 16 + 4 + 1 | 77 = 64 + 9 + 4 |
| 10 = 9 + 1 | 22 = 16 + 4 + 1 + 1 | 87 = 81 + 4 + 1 + 1 |
| 11 = 9 + 1 + 1 | 23 = 9 + 9 + 4 + 1 | 92 = 81 + 9 + 1 + 1 |
| 12 = 9 + 1 + 1 + 1 | 24 = 16 + 4 + 4 | 114 = 100 + 9 + 4 + 1 |
1. Which of the integers from 10 to 20 have more than one representation when expressed
as the sum of four or fewer squares?
Of the integers from 1 to 20, all but 11, 14, and 15 have more than one representation.
2. Attempt to write each of the integers from 1 to 20 as a sum of three or fewer squares.
For which of these numbers can this not be done?
Of the integers from 1 to 20, only 7 and 15 cannot be expressed as the sum of three
or fewer squares.
3. Express the year you were born as a sum of four or fewer squares.
1946 = 1936 + 9 + 1
1946 = 1849 + 81 + 16