Question 3
Twelve students in a class are each given a square card. The side length of each card is a whole number of centimetres from $1$ to $12$ and no two cards are the same size. Each student cuts his/her card into unit squares (of side length $1$ cm). The teacher challenges them to join all their unit squares edge to edge to form a single larger square without gaps. They find that this is impossible.
Alice, one of the students, originally had a card of side length $a$ cm. She says, ‘If I don’t use any of my squares, but everyone else uses their squares, then it is possible!’
Bob, another student, originally had a card of side length $b$ cm. He says, ‘Me too! If I don’t use any of my squares, but everyone else uses theirs, then it is possible!’
Assuming Alice and Bob are correct, what is $ab$?
Alice, one of the students, originally had a card of side length $a$ cm. She says, ‘If I don’t use any of my squares, but everyone else uses their squares, then it is possible!’
Bob, another student, originally had a card of side length $b$ cm. He says, ‘Me too! If I don’t use any of my squares, but everyone else uses theirs, then it is possible!’
Assuming Alice and Bob are correct, what is $ab$?