# Puzzles

## Archive

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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

List of all puzzles

## Tags

arrows number games prime numbers perimeter clocks remainders graphs books probabilty colouring cube numbers folding tube maps coins mean star numbers square roots sport area dice planes floors cards rectangles angles cryptic clues wordplay differentiation scales polygons logic volume fractions crossnumbers surds unit fractions addition parabolas cryptic crossnumbers sum to infinity shape christmas 3d shapes partitions crosswords advent ellipses odd numbers complex numbers balancing multiples triangle numbers ave quadratics factorials geometry people maths factors integration chocolate calculus dodecagons squares indices triangles grids 2d shapes probability perfect numbers speed division chess spheres sums bases pascal's triangle numbers shapes palindromes digits hexagons trigonometry menace chalkdust crossnumber symmetry dates money doubling averages time multiplication coordinates irreducible numbers proportion algebra sequences means rugby circles regular shapes functions square numbers lines percentages routes integers taxicab geometry## 22 December

In base 2, 1/24 is
0.0000101010101010101010101010...

In base 3, 1/24 is
0.0010101010101010101010101010...

In base 4, 1/24 is
0.0022222222222222222222222222...

In base 5, 1/24 is
0.0101010101010101010101010101...

In base 6, 1/24 is
0.013.

Therefore base 6 is the lowest base in which 1/24 has a finite number of digits.

Today's number is the smallest base in which 1/10890 has a finite number of digits.

Note: 1/24 always represents 1 divided by twenty-four (ie the 24 is written in decimal).

## 121

Find a number base other than 10 in which 121 is a perfect square.

## Adding bases

Let \(a_b\) denote \(a\) in base \(b\).

Find bases \(A\), \(B\) and \(C\) less than 10 such that \(12_A+34_B=56_C\).

## Reverse bases again

Find three digits \(a\), \(b\) and \(c\) such that \(abc\) in base 10 is equal to \(cba\) in base 9?

## Reverse bases

Find two digits \(a\) and \(b\) such that \(ab\) in base 10 is equal to \(ba\) in base 4.

Find two digits \(c\) and \(d\) such that \(cd\) in base 10 is equal to \(dc\) in base 7.

Find two digits \(e\) and \(f\) such that \(ef\) in base 9 is equal to \(fe\) in base 5.