On a sheet of paper, 100 characters fit in width and 100 in height. On this sheet, the numbers $1, 2, 3$, and so on, are written successively, with a space between each one and the next. If at the end of a line there is not enough space to write the next number, it is written at the beginning of the next line. What is the last number written on the sheet?
On the 1$^{\text {st }}$ line, we write the numbers from 1 to 9, each followed by a space, which occupies 18 spaces; 82 spaces remain. Each two-digit number plus one space occupies three places on the line. Since $82=27 \times 3+1$, we complete the 1st line with 27 two-digit numbers, starting from the number 10. Thus, the last number of the first line is 36, and one space remains. Denoting each space between numbers by a dash, we can represent the 1st line as follows.

$$
1^{\underline{st}} \operatorname{line}: \underbrace{1-2-3-4-5-6-7-8-9}_{18} \underbrace{10-\cdots-36-}_{81}
$$

But $100=33 \times 3+1$, so we can place 33 two-digit numbers on the second line, each followed by its respective space, and one space remains at the end of the line.

$$
2^{\mathrm{nd}} \text { line }: \underbrace{37-38-\cdots-69-}_{99}-
$$

On the 3rd line, we place the numbers from 70 to 99, occupying $30 \times 3=90$ spaces. We use the remaining ten spaces for the first two three-digit numbers, each followed by a space; at the end of the line, two spaces remain.

$$
3^{\underline{rd}} \text { line : } \underbrace{70-71-\cdots-99}_{90} \underbrace{100-101-}_{8}--
$$

Starting from the fourth line, we can place $100 \div 4=25$ three-digit numbers with their respective spaces in each line. From 102 to 999, inclusive, we have $999-102+1=$ 898 numbers. Since $898=25 \times 35+23$, we occupy 35 lines, from the 4th to the 38th, with these three-digit numbers, and 23 of these numbers remain, which we place on the 39th line, where they occupy $23 \times 4=92$ spaces, leaving eight spaces, which we fill with 1000, leaving three.

$$
39^{\mathrm{th}} \text { line : } \underbrace{977-\cdots-999-}_{92} \underbrace{1000-}_{5}---
$$

Starting from the 40th line, we can place $100 \div 5=20$ four-digit numbers with their respective spaces in each line. Thus, in the remaining 61 lines, we can place exactly $61 \times 20=1220$ four-digit numbers. Starting with 1001, we end the 100th line with the number 2220.