On January 1st, 2023, a person saved ` 1. On January 2nd, 2023, he saved ` 2 more than that on the previous day. On January 3rd, 2023, he saved ` 2 more than that on the previous day and so on. At the end of which date was his total savings a perfect square as well a perfect cube?

The correct option is 8th January, 2023.

Let us analyze the savings day by day to understand the pattern. The problem states that:
- On January 1st (Day 1), the person saved $1$ rupee.
- On January 2nd (Day 2), he saved $2$ more than on the previous day. So, he saved $1+2=3$ rupees.
- On January 3rd (Day 3), he saved $2$ more than on the previous day. So, he saved $3+2=5$ rupees.
- This pattern continues, meaning the savings on each day form an arithmetic progression of odd numbers: $1,3,5,7,...$

The sum of the first $n$ odd natural numbers is well-known to be a perfect square, specifically $n^2$. Let's verify this step-by-step:
- At the end of Day 1: Total savings = $1=1^2$
- At the end of Day 2: Total savings = $1+3=4=2^2$
- At the end of Day 3: Total savings = $1+3+5=9=3^2$
- Generally, at the end of Day $n$ (where $n$ represents the date in January), the total savings will be $n^2$ rupees.

We are looking for a date $n$ such that the total savings, $n^2$, is both a perfect square and a perfect cube.
For a number to be both a perfect square and a perfect cube, it must be of the form $k^6$ for some positive integer $k$.
Thus, we set:

$n^2=k^6$

Taking the square root on both sides:

$n=k^3$

This means that the day number $n$ itself must be a perfect cube. Let's look at the smallest perfect cubes:
- For $k=1$, $n=1^3=1$ (January 1st). The savings would be $1^2=1$, which is both a square and a cube. However, this is not in the options.
- For $k=2$, $n=2^3=8$ (January 8th). The savings at the end of the 8th day would be:

$8^2=64$

Let's check if $64$ is a perfect square and a perfect cube:
- It is a perfect square: $8^2=64$
- It is a perfect cube: $4^3=64$
Since $64$ satisfies both conditions, the date is indeed the 8th of January, 2023.