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 *x*^{2} – 21 and *x*^{2} + 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 *a*^{2} + *b*^{2} = *c*^{2} and ordered triples (*p*, *q*,* r*) with *p*^{2}, *q*^{2}, *r*^{2} 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.