KA-0134Paper folding and hole punching
5 points, difficulty 3 of 3Level 3–4about 105sA square sheet is folded in half, then in half again. Two holes are punched through all the layers. How many holes are there when it is unfolded?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count the layers first, then the holes.
2The strategy
Each fold doubles the number of layers. Work out how many layers the punch goes through, then multiply by how many holes were punched.
3The full solution
Two folds make 2 x 2 = 4 layers, and each of the 2 holes goes through all of them: 4 x 2 = 8.
Solution
The reliable way
Count the layers. One fold gives 2 layers, a second fold gives 4. Each punch goes through all 4 layers, leaving 4 holes in the unfolded sheet. With 2 punches that is 4 x 2 = 8 holes. The transferable idea: layers first, then holes - the two numbers multiply, they do not add.
The elegant way
n folds give 2 to the power n layers, so h holes punched through become h times 2 to the power n: 2 x 4 = 8.
Why this is on the test: Folding questions combine a doubling and a multiplication, and mixing up which is which produces every wrong answer offered.