IBM Ponder This #304 · August 2023

This puzzle was suggested by Michael Brand - thanks Michael!

A square number is a number of the form $n^2$ for some $n$.

A triangular number is a number of the form $1+2+3+\dots +m$ for some m.

The number 1225 is both square and triangular: $1225={35}^2=1+2+3+\dots +49$. Such a number is called square-triangular.

It is easy to see that there are exactly 7 positive square-triangular numbers below ${10}^{10}$.

Your goal: Find the number of positive square-triangular numbers below ${10}^{100}$ (googol).

A bonus "*" will be given for finding the number of positive square-triangular numbers below ${10}^{{10}^{100}}$ (googolplex).

A "**" will be given for finding the exact solution to the "*" part; a good approximation (5 significant digits) is enough for a "*"

Solution

Best opened after a real attempt

To be added.