mscroggs.co.uk
mscroggs.co.uk

subscribe

Puzzles

8 December

The residents of Octingham have 8 fingers. Instead of counting in base ten, they count in base eight: the digits of their numbers represent ones, eights, sixty-fours, two-hundred-and-fifty-sixes, etc instead of ones, tens, hundreds, thousands, etc.
For example, a residents of Octingham would say 12, 22 and 52 instead of our usual numbers 10, 18 and 42.
Today's number is what a resident of Octingham would call 11 squared (where the 11 is also written using the Octingham number system).

Show answer

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 2020

Advent calendar 2019

Sunday Afternoon Maths LXVII

Coloured weights
Not Roman numerals

Advent calendar 2018


List of all puzzles

Tags

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

Archive

Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2021