185. Squares and triangular numbers. What is the third largest number (the smallest number is considered the first) that is both a triangular number ${ }^{*}$ and a square? Of course, the first two numbers with this property are 1 and 36. What is the next number?
185. To find a number that is both a square and a triangular number, one must solve the Pell equation: \(8 x^{2}+1=y^{2}\). Successive values for \(x\) are \(1,6,35\), etc., and for \(y\) are \(3,17,99\), etc. The answer is the number 1225 \(\left(35^{2}\right)\), which has the required properties.