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

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

Archive

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