# Puzzles

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#### Sunday Afternoon Maths LXVII

Coloured weightsNot Roman numerals

#### Advent calendar 2018

#### Sunday Afternoon Maths LXVI

Cryptic crossnumber #2#### Sunday Afternoon Maths LXV

Cryptic crossnumber #1Breaking Chocolate

Square and cube endings

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grids perimeter algebra rectangles scales sport folding tube maps money christmas factorials circles proportion irreducible numbers integers ellipses factors sums unit fractions cryptic clues crossnumbers geometry square roots remainders indices calculus floors triangle numbers percentages 2d shapes shape books cube numbers taxicab geometry dice logic quadratics multiples symmetry speed differentiation volume means prime numbers chalkdust crossnumber palindromes numbers polygons routes partitions mean trigonometry dodecagons rugby cards chocolate integration people maths dates bases multiplication arrows chess cryptic crossnumbers squares lines hexagons ave parabolas 3d shapes odd numbers averages perfect numbers probability digits sequences area surds menace planes triangles wordplay functions probabilty shapes time star numbers angles division coordinates clocks sum to infinity complex numbers square numbers coins addition crosswords doubling games graphs number balancing regular shapes pascal's triangle spheres advent fractions colouring## 23 December

This number is a prime number. If you treble it and add 16, the result is also prime. Repeating this will give 11 prime numbers in total (including the number itself).

## 14 December

What is the only palindromic three digit prime number which is also palindromic when written in binary?

## 3n+1

Let \(S=\{3n+1:n\in\mathbb{N}\}\) be the set of numbers one more than a multiple of three.

**(i)**Show that \(S\) is closed under multiplication.

ie. Show that if \(a,b\in S\) then \(a\times b\in S\).

Let \(p\in S\) be irreducible if \(p\not=1\) and the only factors of \(p\) in \(S\) are \(1\) and \(p\). (This is equivalent to the most commonly given definition of prime.)

**(ii)**Can each number in \(S\) be uniquely factorised into irreducibles?