mscroggs.co.uk
mscroggs.co.uk

subscribe

Puzzles

23 December

I draw the parabola \(y=x^2\) and mark points on the parabola at \(x=17\) and \(x=-6\). I then draw a straight line connecting these two points.
At which value of \(y\) does this line intercept the \(y\)-axis?

Show answer

19 December

The equation \(352x^3-528x^2+90=0\) has three distinct real-valued solutions.
Today's number is the number of integers \(a\) such that the equation \(352x^3-528x^2+a=0\) has three distinct real-valued solutions.

Show answer

Tags: graphs, cubics

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

Archive

Show me a random puzzle
 Most recent collections 

Advent calendar 2023

Advent calendar 2022

Advent calendar 2021

Advent calendar 2020


List of all puzzles

Tags

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

Archive

Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2024