Cubes
4/27/2000 Version
http://www.cs.ucla.edu/~klinger/findcube.html
In the first century A.D. the question
How can the cubes be represented in terms of the natural numbers?
was answered by the statement:
Cubical numbers are always equal to the sum of successive odd numbers and can be represented this way.
A few examples show what this means:
1
^{3}
= 1 = 1
2
^{3}
= 8 = 3 + 5
3
^{3}
= 7 + 9 + 11
4
^{3}
= 13 + 15 + 17 + 19
Determine why this is so. To check your understanding, find the odd decomposition of 7
^{3}
.