IBM Ponder This #308 · December 2023

This riddle was proposed by Hugo Pfoertner - thanks Hugo!

We consider the infinite grid of equilateral triangles with sides length 1, such that $(0,0)$ is a grid point.

In this grid, we consider all circles passing through at least one grid point. The smallest circle has radius 1, hitting the grid point $(1,0)$ and 5 other grid points.

We are interested in all the circles with non-integer radii. In the following illustration, all the circles are drawn, and a blue mark denotes the intersection point of a circle with a non-integer radius along the positive direction of the x-axis.

For every positive integer $m$, $f(m)$ denotes the number of circles that pass through at least one grid point, with a radius of $x$ such that $m<x<m+1$.

For example, $f(11)=5$ and $f(42)=19$.

Your goal: Find a value of $m$ such that $f(m)=1,000,000$.

Hint: There is such an $m$ which is a base 10 palindrome, but other solutions are accepted as well.

A Bonus "*" will be given for finding **all** the values of $m$ such that $f(m)=1,000,000$.

Solution

Best opened after a real attempt

To be added.