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

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

Archive

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