IBM Ponder This #320 · December 2024

Denote by ${\omega }_d(n)$ the size of the set of all natural numbers between $1$ and $n$ that have the digit $d$ in their usual decimal representation. For example, ${\omega }_1(1456)=728$ and ${\omega }_7(1456)=352$.

We see that for $n=1456$, we have $\frac{{\omega }_1(n)}{n}=\frac{1}{2}$, i.e., exactly 50% of the numbers up to 1456 contain the digit 1. For $d=7$, we have $\frac{{\omega }_7(n)}{n}=\frac{22}{91}$.

Your goal: For each $d$ between 1 and 9, find the maximum value of $n$ for which $\frac{{\omega }_d(n)}{n}=\frac{1}{2}$.

An optional goal: Find a simple formula $f(d)$ giving those maximum values for as many values of $d$ as possible.

A bonus "*" will be given for finding the maximum value of $n$ for which $\frac{{\omega }_1(n)}{n}=\frac{3}{4}$ (i.e., only solve the case $d=1$. but for $\frac{3}{4}$ this time).

Solution

Best opened after a real attempt

To be added.