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Source: Maths Jam
Can you order the integers 1 to 16 so that every pair of adjacent numbers adds to a square number?
For which other numbers \(n\) is it possible to order the integers 1 to \(n\) in such a way?
Source: Woody at Maths Jam
Multiply together the first 100 factorials:$$1!\times2!\times3!\times...\times100!$$
Find a number, \(n\), such that dividing this product by \(n!\) produces a square number.
Lots of Ones
Is any of the numbers 11, 111, 1111, 11111, ... a square number?
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.