# Puzzles

## Archive

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#### Sunday Afternoon Maths LIX

Turning SquaresElastic Numbers

Square Pairs

#### Sunday Afternoon Maths LVIII

Factorial PatternPlacing Plates

#### Advent Calendar 2016

#### Sunday Afternoon Maths LVII

Largest Odd FactorsList of All Puzzles

## Tags

time geometry 2d shapes 3d shapes numbers spheres trigonometry complex numbers algebra lines graphs coordinates odd numbers fractions differentiation calculus folding tube maps ellipses triangle numbers money bases triangles squares area square numbers chess probability circles averages speed sport multiples dates factors parabolas functions logic cards games people maths shape prime numbers irreducible numbers probabilty angles proportion dice integration sum to infinity dodecagons hexagons multiplication factorials coins shapes regular shapes colouring grids floors integers rugby crosswords percentages digits sums rectangles clocks menace routes taxicab geometry remainders chalkdust crossnumber palindromes sequences means unit fractions division square roots surds doubling quadratics indices symmetry planes volume number partitions ave pascal's triangle mean advent arrows## Square Pairs

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?

## Square Factorials

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?

## 22 December

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.

## Odd Squares

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.

## 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.