Counting the holes

The idea

Each remaining square makes a new hole at the next step and old holes stay, so Hn+1 = Hn + Rn, giving H1 = 1, H2 = 1 + 8, H3 = 1 + 8 + 8², and so on.

Holes in the carpet

The carpet has holes too. Step 1 has 1 hole, the middle square we took out.

At step 2, every square that remains gets a new hole in its middle. The old hole is still there.

How many holes does step 2 have in all?

• 8
• 9
• 64

Count the new and the old holes

Press Next step. The grey holes are old. The orange holes are new. One square of the step before is outlined: it made one new hole.

Old holes stay, new holes are added

Each remaining square makes a new hole at the next step, and the old holes stay, so Hₙ₊₁ = Hₙ + Rₙ.

H₁ = 1 H₂ = 1 + 8 H₃ = 1 + 8 + 8² and so on.

The new holes at a step are as many as the squares that remained at the step before: 1, 8, 8², …

Notes

The old holes stay and each remaining square makes one new hole: Hₙ₊₁ = Hₙ + Rₙ. So H₁ = 1, H₂ = 1 + 8, H₃ = 1 + 8 + 8², and so on.

Check yourself

Step 2 has 9 holes. Step 2 has 64 remaining squares. How many holes does step 3 have?

Answer: 73 holes

H₃ = H₂ + R₂ = 9 + 64 = 73. Check with the pattern: 1 + 8 + 64 = 73.

Step 3 has 73 holes and 512 remaining squares. How many holes does step 4 have?

Answer: 585 holes

H₄ = 73 + 512 = 585. Same as 1 + 8 + 64 + 512 = 585.

How many new holes are made at step 3?

Answer: 64 holes

Step 2 left 8² = 64 squares. Each makes one new hole at step 3, so 64 new holes.

Going from one step to the next, which is true?

• Every hole splits into 8 holes. Holes do not split. They just stay as they are.
• The old holes are filled in and new holes are made. Nothing is ever filled in. The old holes stay.
• The old holes stay and each remaining square makes one new hole — correct. Yes! That is Hₙ₊₁ = Hₙ + Rₙ.