mscroggs.co.uk
mscroggs.co.uk

subscribe

Puzzles

1 December

One of the vertices of a rectangle is at the point \((9, 0)\). The \(x\)-axis and \(y\)-axis are both lines of symmetry of the rectangle.
What is the area of the rectangle?

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 \(p(6)\)?

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

Balanced sets

A set of points in the plane is called 'balanced' if for any two points \(A\) and \(B\) in the set, there is another point \(C\) in the set such that \(AC=BC\) (here \(AC\) is the distance between \(A\) and \(C\)).
For all \(n\geq3\), find a balanced set of \(n\) points.

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 2023

Advent calendar 2022

Advent calendar 2021

Advent calendar 2020


List of all puzzles

Tags

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

Archive

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