, 2024

Each Tiny three-digit integer belongs to exactly one of the
following three cases.

Case 1: The units digit is $0$ If the units digit of a Tiny integer is $0$, then the tens digit must also be $0$, otherwise, the units digit and tens digit can be switched to give a smaller integer. In this case, there are no restrictions on the hundreds digit and thus there are 9 such Tiny integers. These are: $100$, $200$, $300$, $400$, $500$, $600$, $700$, $800$, $900$. Case 2: The units digit is not $0$, but the tens digit is $0$ If the hundreds digit is $x$ and the units digit is $z$, then the integers in this case are of the form $x0z$, where $z\neq0$. (If $x$ is greater than $z$, then switching $x$ and $z$ creates a smaller integer.) Integers of this form are Tiny exactly when $x$ is greater than or equal to $1$, and $x$ is less than or equal to $z$. If $x=1$, then $z$ can be equal to any integer from $1$ to $9$ inclusive, and so there are $9$ such Tiny integers. These are: $101,102, 103, \dots, 108,109$. If $x=2$, then $z$ can be equal to any integer from $2$ to $9$ inclusive, and so there are $8$ such Tiny integers. These are: $202, 203, 204, \dots, 208,209$. Continuing in this way, there are $7$ Tiny integers when $x=3$, $6$ when $x=4$, $5$ when $x=5$, $4$ when $x=6$, $3$ when $x=7$, 2 when $x=8$, and finally $1$ when $x=9$. In this case, there are $9+8+7+6+5+4+3+2+1=45$ Tiny integers. Case 3: The units digit and the tens digit are both not $0$ If the hundreds digit is $x$ (where $x$ is greater than or equal to $1$), the tens digit is $y$, and the units digit is $z$, then the integers in this case are of the form $xyz$. Integers of this form are Tiny exactly when $x$ is less than or equal to $y$, and $y$ is less than or equal to $z$. For $x=1$, we count the number of such Tiny integers in the table that follows. Value of $x$ Value of $y$ Possible values of $z$ Number of Tiny integers $x=1$ $y=1$ $z=1,2,3,4,\dots,9$ $9$ $x=1$ $y=2$ $z=2,3,4,\dots,9$ $8$ $x=1$ $y=3$ $z=3,4,\dots,9$ $7$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $x=1$ $y=8$ $z=8,9$ $2$ $x=1$ $y=9$ $z=9$ $1$ When $x=1$, there are $9+8+7+6+5+4+3+2+1=45$ Tiny integers in this case. For $x=2$, we may similarly count the number of Tiny integers. Value of $x$ Value of $y$ Possible values of $z$ Number of Tiny integers $x=2$ $y=2$ $z=2,3,4,\dots,9$ $8$ $x=2$ $y=3$ $z=3,4,\dots,9$ $7$ $x=2$ $y=4$ $z=4,\dots,9$ $6$ ⋮ ⋮ ⋮ ⋮ $x=2$ $y=8$ $z=8,9$ $2$ $x=2$ $y=9$ $z=9$ $1$ When $x=2$, there are $8+7+6+5+4+3+2+1=36$ Tiny integers in this case. Notice that for each increase in the value of $x$ by $1$, the smallest possible value of $y$ increases by $1$ (to match the value of $x$), and so the smallest possible value of $z$ also increases by $1$ (to match the value of $y$). This means that when $x=3$, for example, the number of Tiny integers in the first row of the corresponding table is 1 less than the first row of the table for $x=2$, and thus is 7. That is, when $x=3$, there are $7+6+5+4+3+2+1=28$ Tiny integers, and when $x=4$, there are $6+5+4+3+2+1=21$ Tiny integers. Continuing in this way, we summarize the count of Tiny integers for Case 3. Value of $x$ Number of Tiny integers $x=1$ $9+8+7+6+5+4+3+2+1=45$ $x=2$ $8+7+6+5+4+3+2+1=36$ $x=3$ $7+6+5+4+3+2+1=28$ $x=4$ $6+5+4+3+2+1=21$ $x=5$ $5+4+3+2+1=15$ $x=6$ $4+3+2+1=10$ $x=7$ $3+2+1=6$ $x=8$ $2+1=3$ $x=9$ $1$ The number of Tiny three-digit integers in this case is $$45+36+28+21+15+10+6+3+1=165$$ and so the total number of Tiny three-digit integers is $9+45+165=219$.