1. Find all the ways to express 225 as a sum of consecutive odd integers. Use your results to find the squares that can be added to 225 to produce another square. What determines the number of ways in which a given number can be expressed as a sum of consecutive odd numbers?
2. Show that 336 is a congruous number. Use your results to find a rational number x such that x2 – 21 and x2 + 21 are both squares of rational numbers. Can you find examples with numbers other than 5 (shown in the text) and 21?
3. There is a correspondence between ordered triples (a, b, c) with a2 + b2 = c2 and ordered triples (p, q, r) with p2, q2, r2 forming an arithmetic progression. The triple (a, b, c) = (3, 4, 5) corresponds to (p, q, r) = (1, 5, 7), the triple (a, b, c) = (5, 12, 13) corresponds to (p, q, r) = (7, 13, 17), and the triple (a, b, c) = (8, 15, 17) corresponds to (p, q, r) = (7, 17, 23).
Discover the rule for this correspondence and explain why it works.
4. Triangular numbers can be found by the taking the sum of all integers from 1 to n, so we get 1 = 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10, and so on. Adapt as many of Leonardo’s results as you can to the case of triangular numbers.