How many squares remain?
Every remaining square gives 8 new ones.
Counting the squares
Let Rₙ be the number of squares that remain at step n. At step 0 there is just 1 square. At step 1 there are 8.
How many squares remain at step 2?
What this lesson covers
The idea
Every remaining square gives rise to 8 remaining squares at the next step, so Rn+1 = 8Rn and, with R0 = 1, the number of remaining squares at the nth step is Rn = 8ⁿ.
Counting the squares
Let Rₙ be the number of squares that remain at step n. At step 0 there is just 1 square. At step 1 there are 8.
How many squares remain at step 2?
- 16
- 64
- 72
Watch the count grow
Press Next step and watch the table. The orange squares all came from one square of the step before. Tap any square to see where its group came from.
Multiply by 8, every step
Every remaining square gives rise to 8 remaining squares at the next step, so Rₙ₊₁ = 8 Rₙ. With R₀ = 1, the number of remaining squares at the nth step is Rₙ = 8ⁿ.
R₀ = 1 R₁ = 8 × 1 = 8 R₂ = 8 × 8 = 8² R₃ = 8 × 8 × 8 = 8³
The power of 8 is the step number. Step 3 has 8³ = 512 squares.
Notes
Every remaining square gives 8 remaining squares: Rₙ₊₁ = 8 Rₙ. With R₀ = 1, the number of remaining squares at step n is Rₙ = 8ⁿ.
Check yourself
Step 3 of the carpet has 512 remaining squares. How many remain at step 4?
Answer: 4096 squares
R₄ = 8 × R₃ = 8 × 512 = 4096. And 8⁴ = 4096.
Which formula gives the number of remaining squares at step n?
Use the formula to find how many squares remain at step 5.
Answer: 32768 squares
R₅ = 8⁵ = 8 × 4096 = 32768.
Each remaining square gives rise to 8 remaining squares at the next step, not 9. Why?
- Rₙ = 8 × n. That adds 8 each step: 8, 16, 24. But the count is multiplied by 8 each step: 8, 64, 512.
- Rₙ = n⁸. That has the base and the exponent the wrong way round. At step 2, 2⁸ = 256, but the carpet has 64.
- Rₙ = 8ⁿ — correct. Yes! Start with 1 and multiply by 8 at every step: 8 multiplied n times.
- Only the corners remain. The four middle-of-the-side squares remain too. Only the centre is removed.
- The middle square of the 9 is removed — correct. Yes! A square is broken into 9, and the central one is taken out, so 8 are left.
- One square is lost at every step. It is always the same: from each square, exactly the central one of the nine is removed.