The following number is formed by starting from 1 after the decimal point and writing down the next square number after each square number.

$$
0.149162536 \ldots
$$

What will be the 100th, the 1000th, and the 10,000th decimal digit of the number?
Two consecutive square numbers either have the same number of digits, or the larger one has 1 more digit. It is easy to see that the latter type of pair is formed by the square number preceding $10^{2}, 100^{2}, 1000^{2}$, because these are the smallest 3-, 5-, and 7-digit (integer) numbers, respectively. Furthermore, from the square table, it can be read that the smallest 2-, 4-, and 6-digit square numbers are $4^{2}$, $32^{2}$, and $317^{2}$, respectively. From these, we prepare the following table for the answer:

| How many digits | From | To | How many | The number of digits used in this group | Total number of digits used so far |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | $1^{2}$ | $3^{2}$ | 3 | 3 | 3 |
| 2 | $4^{2}$ | $9^{2}$ | 6 | 12 | 15 |
| 3 | $10^{2}$ | $31^{2}$ | 22 | 66 | 81 |
| 4 | $32^{2}$ | $99^{2}$ | 68 | 272 | 353 |
| 5 | $100^{2}$ | $316^{2}$ | 217 | 1085 | 1438 |
| 6 | $317^{2}$ | $999^{2}$ | 683 | 4098 | 5536 |
| 7 | $1000^{2}$ |  |  |  |  |

We see that the 100th digit is in the 19th position of the 4-digit numbers, which is $100-81=19$. So far, we have written $19: 4=4$ complete square numbers, from $32^{2}$ to $35^{2}$, and since the remainder of the division is 3, the third digit of $36^{2}=1296$ is 9, which goes to the 100th position.

Similarly, the 1000th digit, starting from the beginning of the 5-digit square numbers, is the $1000-353=647$th, $647=5 \cdot 129+2$, so we need to determine the 2nd digit of the 130th 5-digit square. The base number is $99+130=229$, and the 2nd digit of its square can be determined even from the rounded data in the table: 2 (since $229^{2}=52441$).

Similarly, $10000-5536=4464=7 \cdot 637+5$, and $(999+638)^{2}=1637^{2}=2679769$ has 7 as its 5th digit. According to all of this, the 100th, 1000th, and 10000th digits of the (1) number are 9, 2, and 7, respectively.

Remark. The digits of the (1) number do not show any regularity, the number is irrational. A more famous example of this is if we write the digits of the consecutive base numbers, the natural numbers, in the place of the consecutive decimal digits, instead of the squares.