What is the largest number which cannot be written as the sum of distinct squares?
Products and sums of squares
Show that the product of any two numbers, each of which is the sum of two square integers, is itself the sum of two square integers.
Source: Maths Jam
Prove that 1 and 9 are the only square numbers where all the digits are odd.
Triangles between squares
Prove that there are never more than two triangle numbers between two consecutive square numbers.
Source: Lewis Carroll's Games & Puzzles
Towards the end of his life, Lewis Carroll recorded in his diary that he had discovered that double the sum of two square numbers could always be written as the sum of two square numbers. For example$$2(3^2 +4^2 )=1^2 +7^2$$ $$2(5^2 +8^2 )=3^2 +13^2$$
Prove that this can be done for any two square numbers.
It was once claimed that there are 204 squares on a chessboard. Can you justify this claim?