#math trivia for #April27: #118 is the sum of three squares (not necessarily distinct) in two different ways. What are they?

— Burt Kaliski Jr. (@modulomathy) April 27, 2012

The two ways are:

118 = 1+36+81

118 = 9+9+100

One way to find the answers is to start by observing that at least one of the three squares must be less than or equal to one third of the sum, which means that the largest value it can have is 36. The second of the three squares must then be less than or equal to one half of the remainder. This leads to the following logic:

* Smallest 1; next can be 1, 4, 9, 16, 25, 36, or 49; only (1,36,81) works

* Smallest 4; next can be 4, 9, 16, 25, 36, or 49; nothing works

* Smallest 9; next can be 9, 16, 25, 36, or 49; only (9,9,100) works

* Smallest 16; next can be 16, 25, 36; nothing works

* Smallest 25; next can be 25 or 36; nothing works

* Smallest 36; next can be 36; nothing works