Floor‑function bounding with $\lfloor\sqrt{n}\rfloor$

Solution.

We need to count the number of sequences of three transformations that return triangle $T$ to its original position.

Let’s analyze each transformation:

  1. Rotation by $90°$ counterclockwise: $(x,y) \to (-y,x)$
  2. Rotation by $180°$ counterclockwise: $(x,y) \to (-x,-y)$
  3. Rotation by $270°$ counterclockwise: $(x,y) \to (y,-x)$
  4. Reflection across $x$-axis: $(x,y) \to (x,-y)$
  5. Reflection across $y$-axis: $(x,y) \to (-x,y)$

The original triangle $T$ has vertices $(0,0)$, $(4,0)$, and $(4,3)$.

Step 1: Find transformations that return $T$ to itself

Let’s check which single transformations return $T$ to itself:

  • Rotation by $180°$: $(0,0) \to (0,0)$, $(4,0) \to (-4,0)$, $(4,3) \to (-4,-3)$

    • This does NOT return $T$ to itself.

  • Reflection across $x$-axis: $(0,0) \to (0,0)$, $(4,0) \to (4,0)$, $(4,3) \to (4,-3)$

    • This does NOT return $T$ to itself.

  • Reflection across $y$-axis: $(0,0) \to (0,0)$, $(4,0) \to (-4,0)$, $(4,3) \to (-4,3)$

    • This does NOT return $T$ to itself.

  • Rotation by $90°$: $(0,0) \to (0,0)$, $(4,0) \to (0,4)$, $(4,3) \to (-3,4)$

    • This does NOT return $T$ to itself.

  • Rotation by $270°$: $(0,0) \to (0,0)$, $(4,0) \to (0,-4)$, $(4,3) \to (3,-4)$

    • This does NOT return $T$ to itself.

Step 2: Find sequences of two transformations

Let’s check sequences of two transformations:

Identity transformation: The composition of any transformation with its inverse gives the identity.

  • Rotation by $90°$ followed by rotation by $270°$ = Identity
  • Rotation by $180°$ followed by rotation by $180°$ = Identity
  • Reflection across $x$-axis followed by reflection across $x$-axis = Identity
  • Reflection across $y$-axis followed by reflection across $y$-axis = Identity

Step 3: Find sequences of three transformations

For a sequence of three transformations to return $T$ to itself, the composition of all three must be the identity transformation.

Let’s denote the transformations as:

  • $R_{90}$: Rotation by $90°$
  • $R_{180}$: Rotation by $180°$
  • $R_{270}$: Rotation by $270°$
  • $R_x$: Reflection across $x$-axis
  • $R_y$: Reflection across $y$-axis

We need to find all ordered triples $(A, B, C)$ such that $A \circ B \circ C = \text{Identity}$.

This means $A \circ B = C^{-1}$.

Step 4: Count the valid sequences

Since we have 5 transformations and each has an inverse, we need to count the number of ways to choose $A$ and $B$ such that $A \circ B$ is one of the 5 transformations.

For each choice of $A$ (5 choices), there is exactly one choice of $B$ such that $A \circ B$ equals any given transformation.

However, we need to be more systematic. Let’s use the fact that:

  1. Rotations: $R_{90} \circ R_{270} = R_{180} \circ R_{180} = \text{Identity}$
  2. Reflections: $R_x \circ R_x = R_y \circ R_y = \text{Identity}$

For three transformations to compose to identity, we need:

  • Either all three are the same transformation (and it’s an involution)
  • Or the first two compose to the inverse of the third

Let’s count systematically:

Case 1: All three transformations are the same

  • $R_{180} \circ R_{180} \circ R_{180} = R_{180}$ (not identity)
  • $R_x \circ R_x \circ R_x = R_x$ (not identity)
  • $R_y \circ R_y \circ R_y = R_y$ (not identity)

Case 2: First two compose to the inverse of the third

For $R_{90}$:

  • $R_{90} \circ R_{270} \circ R_{180} = \text{Identity} \circ R_{180} = R_{180}$ (not identity)
  • $R_{90} \circ R_{180} \circ R_{270} = R_{270} \circ R_{270} = R_{180}$ (not identity)

Wait, let me recalculate this more carefully.

Actually, let’s use a different approach. We know that:

  • $R_{90} \circ R_{270} = \text{Identity}$
  • $R_{180} \circ R_{180} = \text{Identity}$
  • $R_x \circ R_x = \text{Identity}$
  • $R_y \circ R_y = \text{Identity}$

For three transformations to compose to identity, we need the first two to compose to the inverse of the third.

The inverses are:

  • $R_{90}^{-1} = R_{270}$
  • $R_{180}^{-1} = R_{180}$
  • $R_{270}^{-1} = R_{90}$
  • $R_x^{-1} = R_x$
  • $R_y^{-1} = R_y$

So we need:

  1. $A \circ B = R_{270}$ and $C = R_{90}$
  2. $A \circ B = R_{180}$ and $C = R_{180}$
  3. $A \circ B = R_{90}$ and $C = R_{270}$
  4. $A \circ B = R_x$ and $C = R_x$
  5. $A \circ B = R_y$ and $C = R_y$

For each case, we need to count how many pairs $(A,B)$ satisfy $A \circ B = \text{target}$.

Since we have 5 transformations, for each target transformation, there are exactly 5 pairs $(A,B)$ such that $A \circ B = \text{target}$ (one for each choice of $A$).

Therefore, the total number of valid sequences is $5 \times 5 = 25$.

Answer: $\boxed{25}$