mscroggs.co.uk
mscroggs.co.uk

subscribe

Puzzles

22 December

In bases 3 to 9, the number 112 is: \(11011_3\), \(1300_4\), \(422_5\), \(304_6\), \(220_7\), \(160_8\), and \(134_9\). In bases 3, 4, 6, 8 and 9, these representations contain no digit 2.
There are two 3-digit numbers that contain no 2 in their representations in all the bases between 3 and 9 (inclusive). Today's number is the smaller of these two numbers.

Show answer

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).

Show answer

121

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

Show answer & extension

Tags: numbers, bases

Adding bases

Let \(a_b\) denote \(a\) in base \(b\).
Find bases \(A\), \(B\) and \(C\) less than 10 such that \(12_A+34_B=56_C\).

Show answer & extension

Tags: numbers, bases

Reverse bases again

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

Show answer & extension

Tags: numbers, bases

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.

Show answer & extension

Tags: numbers, bases

Archive

Show me a random puzzle
 Most recent collections 

Advent calendar 2019

Sunday Afternoon Maths LXVII

Coloured weights
Not Roman numerals

Advent calendar 2018

Sunday Afternoon Maths LXVI

Cryptic crossnumber #2

List of all puzzles

Tags

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

Archive

Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2020