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

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

Archive

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