, 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 $9+45+165=219$.