How many different sizes of square fit on a pegboard?
Stretching rubber bands on a 5×5 pegboard, how many squares of different sizes can you make?
- 1First, pin down what counts as “different.” The question says different sizes — same size in a different position counts only once.Inner question: “What exactly am I counting?”
- 2Start with what you’re sure of. Upright squares: side lengths 1, 2, 3 and 4 — 4 sizes.Inner question: “Bank the certain ones first.”
- 3The key step: “Does a square have to sit upright?” Think of one side as “a pegs right, then b pegs up” — and tilted squares appear.Inner question: “Is there a placement I haven’t thought of?”
- 4List every (a, b) systematically. (1,1), (2,1), (2,2) and (3,1) all fit — 4 more tilted sizes, for 8 in total.
- 5Finally, ask: “How do I know I haven’t missed any?” Every size corresponds to one (a, b) pair; once the pairs are all listed, the sizes are too. That sentence is the “reasoning” the assessment is looking for.
Going further in class: if squares in different positions also count as different, the answer jumps to 50 — and now (1,2) and (2,1) must be counted separately. Why? That layer we save for class.
The most important step
How do I know I haven’t missed any?
Getting 8 earns only half the credit — being able to say “why this is all of them” is what the screening is really looking at.
This question trains: systematic enumeration plus explaining why nothing is repeated or missed.