(2) can students show that 8 of a triangular number plus 1 makes a square number? (a result credited to Plutarch (100CE))
the algebraic justification for this is reasonably straightforward
a diagram is probably more (immediately) compelling:
(3) can students show that a triangular number plus 6 times the next triangular number plus the next triangular number is a square number?
using a diagram?
using algebra?
(4) can students show with a diagram that 9 of a triangular number plus 1 makes another triangular number?
the algebraic justification for this is a little more demanding
however, there is a neat associated diagram:
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