Consider the natural numbers from 100 to 999, listed in ascending order on a rectangular grid paper, one digit per cell, without any gaps. Each row on the paper has 25 cells, so if necessary, the digits of a number can be split at the end of a row. - How many rows on the paper do not contain the digit 0? How many do not contain the digit 9? How many columns do not contain the digit 0? Show that each of the digits $1,2,3, \ldots, 8,9$ appears in every column 1[^0]

[^0]:    ${ }^{1}$ We recommend writing out a few initial rows, identifying patterns, and using them further. However, under no circumstances should the entire list be written out! We will not review any work that includes the complete list or entries from it.


The number of numbers to be written is $999-99=900$, the number of their digits is 2700, so the entire writing extends over $2700: 25=108$ lines. Similarly, the first hundred numbers (from 100 to 199) fill exactly 12 lines. Therefore, the digits of the numbers 200, 300, ..., 900 will start at the beginning of the lines, with the hundreds digit of the 100s numbers placed in a separate column; a similar observation applies to the digits of any pair of numbers that differ only in their hundreds digit. Thus, for the first question, it is enough to think about the first 12 lines.

![](https://cdn.mathpix.com/cropped/2024_05_02_63259e1b1ec9ea45f8bdg-1.jpg?height=43&width=900&top_left_y=339&top_left_x=599)
0 8, 1 0 9, 1 1 0, 1 1 1, 1 1 2, 1 1 3, 1 1 4, 1 1 5, 1 1
6,1 1 7, 1 1 8, 1 1 9,1 2 0,1 2 1, 1 2 2, 1 2 3, 1 2 4,

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![](https://cdn.mathpix.com/cropped/2024_05_02_63259e1b1ec9ea45f8bdg-1.jpg?height=41&width=897&top_left_y=462&top_left_x=603)
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3 3, 1 3 4, 1 3 5, 1 3 6,1 3 7, 1 3 8, 1 3 9, 1 4 0, 1 4
1,1 4 2, 1 4 3, 4 4, 4 5, 4 6, 4 7, 4 8, 4 9,
1 5 0, 1 1,. . . . . . . . . .

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a) In our numbers, 0 only appears in the tens and units place. The ten 0s in the tens place fall into the 1st and 2nd lines because 16 of our numbers fit into these two lines, so even the number 115 is in the 2nd line. The 0 in the units place first appears in the number "100" on the 3rd field of the 1st line. The 0s in the numbers 110, 120, 130, 140 follow 10 fields behind, which is 1 full line and 5 fields to the right. Therefore, their 0s fall into the 2nd line at the 8th field, the 3rd line at the 13th field, the 4th line at the 18th field, and the 5th line at the 23rd field. The 0 in the number "150" would fall into the 28th field of the 6th line, but we cannot write there, so it falls into the 3rd field of the 7th line, directly below the 0 in the units place of the "100" number. Thus, on the one hand, there is no 0 in the 6th line, and on the other hand, from the position of the "150" number, we can see that the digits in the units place repeat every 50 fields. Continuing this way, 0 does not appear in every sixth line, a total of \(108: 6=18\) lines.

b) The 9s that appear in the tens place in the numbers from 100 to 199 only stand in the 11th and 12th lines because the first 10 lines fit 250 fields, 83 numbers fit, the 11th line starts with the 8 of the number "183" and, similarly, ends with the 9 of the number "191". The first 9 in the units place (from the number 109) is the 30th digit, in the 5th field of the 2nd line. Similar to the 0s, the 9s in the numbers 119, ..., 149 stand in the 3rd, 4th, 5th, and 6th lines, and thus there is no 9 in the 1st and 7th lines. Similarly, in the 100-number sequences starting with 200, 300, ..., 800, 2 lines do not contain 9s. Clearly, every line in the sequence starting with 900 contains a 9, so the number of lines without 9s is \(8 \cdot 2=16\).

c) In part (a), we saw that the 0s in the units place stand in the 3rd, 8th, 13th, 18th, and 23rd columns. The first 0 in the tens place (from the number 100) stands in the 2nd column, and the 0s in the numbers 101, 102, ..., 107 are in columns with numbers 3 more: \(5, 8, 11, 14, 17, 20, 23\), so the numbers 8 and 23 appear again. Finally, the 0s in the numbers 108 and 109 are written in the 1st and 4th columns. Therefore, the columns containing 0s are \(5+(8-2)+2=13\) columns, and there are no 0s in 12 columns.

d) Based on our previous observations about the hundreds sequences, it is enough to show that the digit 1 appears in every column as a hundreds place digit. The hundreds place digits occupy every 3rd field. The hundreds place digits in the 1st line stand in the 1st, 4th, 10th, ..., 25th columns, i.e., in columns with numbers that give a remainder of 1 when divided by 3. The first 1 in the 2nd line stands in the 3rd column, and the 1 in the 3rd line stands in the 2nd column (from the number 109, or the number 117), so there is a 1 in every 3rd column and every column that gives a remainder of 2 when divided by 3. There is no other possibility in terms of division by 3, so indeed, we find a 1 in every column, and every other non-zero digit as well.```