mscroggs.co.uk
mscroggs.co.uk

subscribe

Puzzles

Find them all

Find all continuous positive functions, \(f\) on \([0,1]\) such that:
$$\int_0^1 f(x) dx=1\\ \mathrm{and }\int_0^1 xf(x) dx=\alpha\\ \mathrm{and }\int_0^1 x^2f(x) dx=\alpha^2$$

Show answer & extension

Odd and even outputs

Let \(g:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}\) be a function.
This means that \(g\) takes two natural number inputs and gives one natural number output. For example if \(g\) is defined by \(g(n,m)=n+m\) then \(g(3,4)=7\) and \(g(10,2)=12\).
The function \(g(n,m)=n+m\) will give an even output if \(n\) and \(m\) are both odd or both even and an odd output if one is odd and the other is even. This could be summarised in the following table:
\(n\)
oddeven
\(m\)oddevenodd
eoddeven
Using only \(+\) and \(\times\), can you construct functions \(g(n,m)\) which give the following output tables:
\(n\)
oddeven
\(m\)oddoddodd
eoddodd
\(n\)
oddeven
\(m\)oddoddodd
eoddeven
\(n\)
oddeven
\(m\)oddoddodd
eevenodd
\(n\)
oddeven
\(m\)oddoddodd
eeveneven
\(n\)
oddeven
\(m\)oddoddeven
eoddodd
\(n\)
oddeven
\(m\)oddoddeven
eoddeven
\(n\)
oddeven
\(m\)oddoddeven
eevenodd
\(n\)
oddeven
\(m\)oddoddeven
eeveneven
\(n\)
oddeven
\(m\)oddevenodd
eoddodd
\(n\)
oddeven
\(m\)oddevenodd
eoddeven
\(n\)
oddeven
\(m\)oddevenodd
eevenodd
\(n\)
oddeven
\(m\)oddevenodd
eeveneven
\(n\)
oddeven
\(m\)oddeveneven
eoddodd
\(n\)
oddeven
\(m\)oddeveneven
eoddeven
\(n\)
oddeven
\(m\)oddeveneven
eevenodd
\(n\)
oddeven
\(m\)oddeveneven
eeveneven

Show answer & extension

Tags: functions

Bézier curve

A Bézier curve is created as follows:
1) A set of points \(P_0\), ..., \(P_n\) are chosen (in the example \(n=4\)).
2) A set of points \(Q_0\), ..., \(Q_{n-1}\) are defined by \(Q_i=t P_{i+1}+(1-t) P_i\) (shown in green).
3) A set of points \(R_0\), ..., \(R_{n-2}\) are defined by \(R_i=t Q_{i+1}+(1-t) Q_i\) (shown in blue).
.
.
.
\(n\)) After repeating the process \(n\) times, there will be one point. The Bézier curve is the path traced by this point at \(t\) varies between 0 and 1.

What is the Cartesian equation of the curve formed when:
$$P_0=\left(0,1\right)$$ $$P_1=\left(0,0\right)$$ $$P_2=\left(1,0\right)$$

Show answer & extension

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

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

Archive

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