Investigate whether it's possible to tile a stair of length n with stairs of length m, with a specific focus on m=3 | Step-by-Step Solution

What We're Solving

We want to figure out if we can perfectly cover a staircase of length n using only smaller staircases of length m (with special focus on m=3). Think of this like a puzzle - can we arrange smaller stair pieces to exactly fill a larger stair shape without gaps or overlaps?

The Approach

This is a beautiful tiling problem that combines geometry with number theory! We need to think about both the area (total number of unit squares) and the shape constraints. Just like fitting puzzle pieces together, both the total area must match AND the pieces must physically fit together properly.

Step-by-Step Solution

Step 1: Understand what we're working with A "stair of length n" has area 1 + 2 + 3 + ... + n = n(n+1)/2 unit squares. A "stair of length m" has area m(m+1)/2 unit squares.

Step 2: Find the necessary condition For tiling to be possible, the areas must match perfectly:

• If we use k smaller stairs of length m, then: k × m(m+1)/2 = n(n+1)/2
• This means: k = n(n+1)/[m(m+1)]
• For this to work, n(n+1) must be divisible by m(m+1)

Step 3: Focus on the special case m=3 When m=3, each small stair has area 3(4)/2 = 6 unit squares. So we need n(n+1) to be divisible by 12.

Let's check small values:

• n=1: area=1 (not divisible by 6) ❌
• n=2: area=3 (not divisible by 6) ❌
• n=3: area=6 (divisible by 6, k=1) ✓
• n=4: area=10 (not divisible by 6) ❌
• n=5: area=15 (not divisible by 6) ❌
• n=8: area=36 (divisible by 6, k=6) - needs checking ✓

Step 4: Check if area condition is sufficient This is where it gets tricky! Just because the areas match doesn't guarantee the shapes fit together. We need to verify that we can actually arrange the pieces.

For n=3, m=3: This works trivially (one stair covers itself).

For larger cases, this becomes a complex geometric puzzle that often requires computer verification or advanced mathematical techniques.

The Answer

• Necessary condition: n(n+1) must be divisible by m(m+1)
• For m=3: n(n+1) must be divisible by 12
• Small solutions exist: n=3 definitely works, and some larger values like n=8 are believed to work
• General answer: This remains an active area of research! Not all cases where the area condition is met actually have valid tilings.

Memory Tip

Remember "Area first, shape second" - the total area must work out perfectly (necessary condition), but then you still need to solve the geometric puzzle of actually fitting the pieces together (sufficient condition). It's like having enough LEGO pieces of the right total size, but still needing to figure out how they actually connect!

This problem beautifully shows how pure mathematics connects to visual, hands-on puzzles. Keep exploring - mathematics is full of these delightful surprises! 🧩