mscroggs.co.uk
mscroggs.co.uk

subscribe

Puzzles

10 December

For all values of \(x\), the function \(f(x)=ax+b\) satisfies
$$8x-8-x^2\leqslant f(x)\leqslant x^2.$$
What is \(f(65)\)?
Edit: The left-hand quadratic originally said \(8-8x-x^2\). This was a typo and has now been corrected.

Show answer

6 December

\(p(x)\) is a quadratic with real coefficients. For all real numbers \(x\),
$$x^2+4x+14\leq p(x)\leq 2x^2+8x+18$$
\(p(2)=34\). What is \(p(6)\)?

Two tangents

Source: Reddit
Find a line which is tangent to the curve \(y=x^4-4x^3\) at 2 points.

Show answer

Between quadratics

Source: Luciano Rila (@DrTrapezio)
\(p(x)\) is a quadratic polynomial with real coefficients. For all real numbers \(x\),
$$x^2-2x+2\leq p(x)\leq 2x^2-4x+3$$
\(p(11)=181\). Find \(p(16)\).

Show answer

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

Parabola

On a graph of \(y=x^2\), two lines are drawn at \(x=a\) and \(x=-b\) (for \(a,b>0\). The points where these lines intersect the parabola are connected.
What is the y-coordinate of the point where this line intersects the y-axis?

Show answer & extension

Two lines

Let A and B be two straight lines such that the gradient of A is the y-intercept of B and the y-intercept of A is the gradient of B (the gradient and y-intercept of A are not the same). What are the co-ordinates of the point where the lines meet?

Show answer & extension

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

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

Archive

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