Category Archives: Interesting Mathematics

March 3, 2025 · 2:51 am

Derangements

From Wikipedia

In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position.

For example, if we have the set ${A, B, C}$ , there are $6$ permutations

$ABC, ACB, BAC, BCA, CAB, CBA$

But only $2$ of them are derangements – $BCA$ and $CAB$

I did a practical question on this here. In that question I used a tree diagram, but there must be a way to determine the number of derangements given $n$ elements.

We know $3$ elements has $2$ derangements, what about $4$ elements? We know there are $4!=24$ permutations

ABCD	BACD	CABD	DABC
ACBD	BADC	CADB	DACB
ACDB	BCAD	CBAD	DBAC
ABDC	BCDA	CBDA	DBCA
ADBC	BDAC	CDAB	DCAB
ADCB	BDCA	CDBA	DCBA

I have highlighted the derangements. when $n=4$ there are $9$ derangements.

Let’s try to generalise.

In how many ways can one element be in its original position?
$\begin{pmatrix}4\\1\end{pmatrix} \times 3!=24$
Choose one element from the 4 to be in its original position, and then arrange the remaining three elements.

So far, we have $D_3=4!-\begin{pmatrix}4\\1\end{pmatrix} \times 3!=0$

Clearly we are counting some arrangements multiple times, for example ABDC has A and B in the correct position, so we need to add all of the arrangements with $2$ elements in their original position.
In how many ways can two elements be in their original position?
$\begin{pmatrix}4\\2\end{pmatrix} \times 2!=12$

So now we have, $D_3=4!-\begin{pmatrix}4\\1\end{pmatrix} \times 3!+\begin{pmatrix}4\\2\end{pmatrix} \times 2!=12$

But once again we have counted some arrangements multiple times, so we need to subtract all of the arrangements with $3$ elements in their original position.
In how many ways can three elements be in their original position?
$\begin{pmatrix}4\\3\end{pmatrix} \times 1!=4$

So now we have, $D_3=4!-\begin{pmatrix}4\\1\end{pmatrix} \times 3!+\begin{pmatrix}4\\2\end{pmatrix} \times 2!-\begin{pmatrix}4\\3\end{pmatrix} \times 1!=8$

We now need to add all of the arrangements with $4$ elements in their original position.
In how many ways can four elements be in their original position?
$\begin{pmatrix}4\\4\end{pmatrix} \times 0!=1$

So now we have, $D_3=4!-\begin{pmatrix}4\\1\end{pmatrix} \times 3!+\begin{pmatrix}4\\2\end{pmatrix} \times 2!-\begin{pmatrix}4\\3\end{pmatrix} \times 1!+\begin{pmatrix}4\\4\end{pmatrix} \times 0!=9$

Hence,

$D_n=\begin{pmatrix}n\\0\end{pmatrix}\times n!-\begin{pmatrix}n\\1\end{pmatrix}\times (n-1)!+\begin{pmatrix}n\\2\end{pmatrix}\times (n-2)!-... \mp \begin{pmatrix}n\\n\end{pmatrix}\times 0!$

Which we can simplify to

$D_n=\Sigma_{i=0}^{n}(-1)^n\begin{pmatrix}n\\i\end{pmatrix}\times(n-i)!$

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Filed under Counting Techniques, Interesting Mathematics, Year 11 Specialist Mathematics

December 16, 2024 · 9:19 am

Interesting Sum

$S=\sum_{n=1}^\infty (tan^{-1}(\frac{2}{n^2}))$ , find $S$ .

I came across this sum in An Imaginary Tale by Nahin and I was fascinated.

Let $tan(\alpha)=n+1$ and $tan(\beta)=n-1$ .

tan(\alpha-\beta)=\frac{tan(\alpha)-tan(\beta)}{1+tan(\alpha)tan(\beta)}

tan(\alpha-\beta)=\frac{(n+1)-(n-1)}{1+(n+1)(n-1)}

Which means,

$\begin{equation*}S=\sum_{n=1}^\infty(tan^{-1}(n+1)-tan^{-1}(n-1))\end{equation}$

Let’s try a few partial sums

$S_4=tan^{-1}(2)-tan^{-1}(0)+tan^{-1}(3)-tan^{-1}(1)+tan^{-1}(4)-tan^{-1}(2)+tan^{-1}(5)-tan^{-1}(3)$

$S_4=-tan^{-1}(0)+-tan^{-1}(1)+tan^{-1}(4)+tan^{-1}(5)$

$S_6=tan^{-1}(2)-tan^{-1}(0)+tan^{-1}(3)-tan^{-1}(1)+tan^{-1}(4)-tan^{-1}(2)+tan^{-1}(5)-tan^{-1}(3)+tan^{-1}(6)-tan^{-1}(4)+tan^{-1}(7)-tan^{-1}(5)$

$S_6=-tan^{-1}(0)+-tan^{-1}(1)+tan^{-1}(6)+tan^{-1}(7)$

Hence, $S_N=-tan^{-1}(0)+-tan^{-1}(1)+tan^{-1}(N)+tan^{-1}(N+1)$

$S_N=-\frac{\pi}{4}-0+tan^{-1}(N)+tan^{-1}(N+1)$

What happens as $N\rightarrow \infty$ ?

$\lim\limits_{N\to \infty}\ S_N=-\frac{\pi}{4}+\frac{\pi}{2}+\frac{\pi}{2}=\frac{3\pi}{4}$

Because we know $tan(\frac{\pi}{2})$ is undefined.

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Filed under Identities, Interesting Mathematics, Puzzles, Sequences, Trigonometry

Tagged as limits, sum, trigonometric identities

December 14, 2024 · 6:56 am

Volume and Surface Area of a Conical Frustrum

My best attempt at drawing a Frustrum in Geogebra.

We have a truncated cone,

(1) $\begin{equation*}V=\frac{1}{3}\pi R_2^2(h_1+h_2)-\frac{1}{3}\pi R_1^2h_1\end{equation*}$

We are unlikely to know $h_1$ . Can we get $h_1$ in terms that we do know (i.e. $R_1, R_2, h_2$ )?

Think of similar triangles

$\Delta ABC \sim \Delta ADE (AA)$

$R_1 \parallel R_2$

$\angle {C}=\angle{E}$ (Corresponding Angles in Parallel Lines)

$\angle {B}=\angle {D}$ (Corresponding Angles in Parallel Lines)

Therefore

$\begin{equation*}\frac{h_1}{R_1}=\frac{h_1+h_2}{R_2}\end{equation}$

Rearrange to make $h_1$ the subject.

$\begin{equation*}h_1=\frac{h_2R_1}{R_2-R_1}\end{equation}$

Substitute into equation $(1)$

$\begin{equation*}V=\frac{1}{3} \pi((R_2^2(\frac{h_2R_1}{R_2-R_1})+h_2)-R_1^2(\frac{h_2R_1}{R_2-R_1}))\end{equation}$

$\begin{equation*}V=\frac{1}{3} \pi (R_2^2h_2+\frac{R_2^2h_2R_1}{R_2-R_1}-\frac{R_1^2h_2R_1}{R_2-R_1})\end{equation}$

$\begin{equation*}V=\frac{1}{3} \pi (R_2^2h_2+\frac{R_2^2h_2R_1-R_1^2h_2R_1}{R_2-R_1})\end{equation}$

$\begin{equation*}V=\frac{1}{3} \pi (R_2^2h_2+\frac{h_2R_1}{R_2-R_1}(R_2^2-R_1^2))\end{equation}$

$\begin{equation*}V=\frac{1}{3} \pi h_2(R_2^2+\frac{R_1}{R_2-R_1}(R_2-R_1)(R_2+R_1))\end{equation}$

$\begin{equation*}V=\frac{1}{3} \pi h_2(R_2^2+R_1(R_2+R_1))\end{equation}$

$\begin{equation*}V=\frac{1}{3} \pi h_2(R_2^2+R_1 R_2+R_1^2)\end{equation}$

Now let’s think about the surface area.

The surface area of a cone is $A=\pi r^2+\pi rs$ where $s$ is the slant height of the cone.

Once again, we need to subtract the ‘missing’ part of the cone.

(2) $\begin{equation*}A=\pi R_2^2+\pi R_2(s_1+s_2)+ \pi R_1^2- \pi R_1s_1\end{equation*}$

We don’t need to subtract the circle of the top cone because it is the top of the frustrum, but we do need to add it on.

Using similar triangles again

$\begin{equation*}\frac{s_1}{R_1}=\frac{s_1+s_2}{R_2}\end{equation}$

$\begin{equation*}s_1=\frac{s_2R_1}{R_2-R_1}\end{equation}$

Substitute into equation $(2)$

$\begin{equation*}A=\pi(R_2^2+R_2(\frac{s_2R_1}{R_2-R_1}+s_2)+\pi R_1^2-R_1(\frac{s_2R_1}{R_2-R_1}))\end{equation}$

$\begin{equation*}A=\pi(R_2^2+R_2s_2+R_1^2+\frac{s_2R_1}{R_2-R_1}(R_2-R_1))\end{equation}$

$\begin{equation*}A=\pi(R_2^2+R_2s_2+R_1s_2+R_1^2)\end{equation}$

$\begin{equation*}A =\pi (R_1^2+s_2(R_1+R_2)+R_2^2)\end{equation}$

And hence the curved surface area is $\pi s_2(R_1+R_2)$ .

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Filed under Area, Area of Frustrum, Interesting Mathematics, Measurement, Volume of Frustrum

Tagged as frustrum, surface area, volume

November 29, 2024 · 8:11 am

Interesting Equation

I think this one is doing the rounds, I first saw it here.

$\begin{equation*}2^x3^{x^2}=6\end{equation}$

$x=1$ is the obvious answer, $2^1\times 3^1=6$ , but are there more answers?

This was my approach

$\begin{equation*}ln(2^x3^{x^2})=ln(6)\end{equation}$

$\begin{equation*}ln(2^x)+ln(3^{x^2})=ln(6)\end{equation}$

$\begin{equation*}xln(2)+x^2ln(3)-ln(6)=0\end{equation}$

$\begin{equation*}ln(3)x^2+ln(2)x-ln(6)=0\end{equation}$

A quadratic equation.

Hence,

$\begin{equation*}x=\frac{-ln(2)\pm\sqrt{(ln(2))^2-4(ln(3))(ln(6))}}{2ln(3)}\end{equation}$

I then used my calculator

Hence $x=1$ 0r $x=-1.631$

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Filed under Algebra, Index Laws, Interesting Mathematics, Quadratics, Solving

Tagged as exponentials, index laws, Interesting maths, quadratic solving

November 26, 2024 · 7:07 am

Infinite Product Expansion of cos (x)

Remember

(1) $\begin{equation*}cos(x)=1-\frac{1}{2!}x^2+\frac{1}{4!}x^4-\frac{1}{6!}x^6+...\end{equation*}$

We know that $cos(x)=0$ for odd integer multiples of $\frac{\pi}{2}$ , i.e. $\frac{\pi}{2}, \frac{3\pi}{2}, ...$ , which is $\frac{(2n-1)\pi}{2}$ for $n\neq 0$

Hence,

$\begin{equation*}0=1-\frac{1}{2!}x^2+\frac{1}{4!}x^4-\frac{1}{6!}x^6+...\end{equation}$

for $x=\frac{(2n-1)\pi}{2}, n>0$

We can factorise our $cos(x)$ expansion

$\begin{equation*}(1-\frac{x^2}{r_1})(1-\frac{x^2}{r_2})...\end{equation}$

We know $r_1=\frac{\pi}{2}, r_2=\frac{3\pi}{2}, ...$

$\begin{equation*}cos(x)=(1-\frac{x^2}{(\frac{\pi}{2})^2})(1-\frac{x^2}{(\frac{3\pi}{2})^2})...(1-\frac{x^2}{(\frac{(2n-1)\pi}{2})^2})\end{equation}$

$\begin{equation*}cos(x)=\Pi_{n=1}^{\infty}(1-\frac{x^2}{(\frac{(2n-1)\pi}{2})^2})\end{equation}$

$\begin{equation*}cos(x)=\Pi_{n=1}^{\infty}(1-\frac{4x^2}{(2n-1)^2\pi^2})\end{equation}$

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Filed under Factorising, Identities, Infinite Product Expansion, Interesting Mathematics, Polynomials, Trigonometry

Tagged as infinite product expansion of cos(x), trigonometry

October 18, 2024 · 7:06 am

Fibonacci Sequence – Finding the Closed Form

I have been reading An Imaginary Tale – The Story of $\sqrt{-1}$ by Paul J Nahin, which is fabulous. There was a bit in chapter 4 where he found the closed form of the generalised Fibonacci sequence. I thought it would be a good exercise to find the closed from of the Fibonacci sequence.

Just to remind you, the Fibonacci sequence is

$1, 1, 2, 3, 5, 8, 13, 21, ...$

and it is defined recursively

$\begin{equation*}T_{n+2}=T_{n+1}+T_n, T_0=1, T_1=1\end{equation}$

That is, the next term is the sum of the two previous terms, i.e.

$\begin{equation*}T_3=T_2+T_1=1+1=2\end{equation}$

Now the starting off point is slightly dodgy as it involves and educated guess as Paul Nahin writes,

How do I know that works? Because I have seen it before, that’s how! […] There is nothing dishonourable about guessing correct solutions – indeed, great mathematicians and scientists, are invariable great guessers – just as long as eventually the guess is verified to work. The next time you encounter a recurrence formula, you can guess the answer too because then you will have already seen how it works.

We start with $T_n=kz^n$

This means $T_{n+2}=T_{n+1}+T_n$ is $kz^{n+2}=kz^{n+1}+kz^n$

$\begin{equation*}kz^{n+2}=kz^{n+1}+kz^n\end{equation}$

$\begin{equation*}kz^n(z^2-z-1)=0\end{equation}$

$\begin{equation*}z^2-z-1=0\end{equation}$

$\begin{equation*}z=\frac{1\pm\sqrt{(-1)^2-4(1)(-1)}}{2}\end{equation}$

$\therefore z=\frac{1+\sqrt{5}}{2}$ or $z=\frac{1-\sqrt{5}}{2}$

Hence $T_n=k_1(\frac{1+\sqrt{5}}{2})^n+k_2(\frac{1-\sqrt{5}}{2})^n$ and we can use the initial conditions $T_0=1$ and $T_1=1$ to find $k_1$ and $k_2$

When $n=0, T_0=1$

(1) $\begin{equation*}1=k_1+k_2\end{equation*}$

When $n=1, T_1=1$

(2) $\begin{equation*}1=k_1(\frac{1+\sqrt{5}}{2})+k_2(\frac{1-\sqrt{5}}{2})\end{equation*}$

From equation $1$ , $k_2=(1-k_1)$ , substitute into equation $2$

$\begin{equation*}1=k_1(\frac{1+\sqrt{5}}{2})+(1-k_1)(\frac{1-\sqrt{5}}{2})\end{equation}$

$\begin{equation*}1=k_1(\frac{1+\sqrt{5}}{2})+\frac{1-\sqrt{5}}{2}-k_1(\frac{1-\sqrt{5}}{2})\end{equation}$

$\begin{equation*}1=k_1\sqrt{5}+\frac{1-\sqrt{5}}{2}\end{equation}$

$\begin{equation*}1-(\frac{1-\sqrt{5}}{2})=\sqrt{5}k_1\end{equation}$

$\begin{equation*}k_1=\frac{1}{\sqrt{5}}(\frac{1}{2}+\frac{\sqrt{5}}{2})\end{equation}$

$\begin{equation*}k_1=\frac{1}{2}(\frac{1}{\sqrt{5}}+1)\end{equation}$

$\begin{equation*}k_1=\frac{1+\sqrt{5}}{2\sqrt{5}}\end{equation}$

$\begin{equation*}k_2=1-\frac{1+\sqrt{5}}{2\sqrt{5}}\end{equation}$

$\begin{equation*}k_2=-(\frac{1-\sqrt{5}}{2\sqrt{5}})\end{equation}$

$\therefore T_n=(\frac{1+\sqrt{5}}{2\sqrt{5}})(\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2\sqrt{5}})(\frac{1-\sqrt{5}}{2})^n$

$T_n=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{n+1}-(\frac{1-\sqrt{5}}{2})^{n+1})$

Does it work?

Remember the sequence is $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...$

If $n=5, T_n=8$

$\begin{equation*}T_5=\frac{1}{\sqrt{5}}(\frac{1+\sqrt{5}}{2})^6-\frac{1-\sqrt{5}}{2})^6)\end{equation}$

As you can see it works!

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Filed under Complex Numbers, Fibonacci, Fibonacci Sequence, Interesting Mathematics, Sequences

May 2, 2024 · 9:54 am

Napoleon’s Theorem

This is one of those theorems, where the name is probably not the person who discovered it. Check out the Wikipedia entry for more information on the history.

Napoleon’s theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral triangle.

I am just going to look at the outward triangles.

Construct equilateral triangles on the edges AB, BC and AC.

Find the centre (Centroid) of these triangles.

Note: The centroid is the point of intersections of the medians.

A median is the line from a vertex to the midpoint of the opposite side.

We are going to prove that triangle PQR is equilateral

I am going to add AP, AQ, QB, BR, RC and CP

As P, Q and R are the centroids, PA=PC, QA=QB, and RB=RC.

Therefore triangles AQB, BRC, and CPA are isosceles.

The angles QAB, QBA, RBC, RCB, PCA, PAC (the shaded angles in the above diagram) are 30 degrees, because the line bisects the angle (and the angle is 60 degrees because the triangles are equilateral).

Now for the maths.

$\textnormal {Let}\ AB=y,\ AC=x,\ BC=z$

$\textnormal {Let}\ \angle{CAB}=\theta,\ \angle{ABC}=\alpha,\ \angle{BCA}=\gamma$

$\therefore\angle{PAQ}=(60+\theta),\ \angle{QBR}=(60+\alpha),$

$\angle{RCP}=(60+\gamma)$

PA is 2/3 the length of the median.

AS is the median

$sin(60)=\frac{AS}{x}$

$AS=xsin(60)$

$AS=\frac{\sqrt{3}}{2}x$

Hence,

$PA=\frac{2}{3}\frac{\sqrt3}{2}x$

$PA=\frac{\sqrt{3}}{3}x$

$PA=\frac{x}{\sqrt{3}}$

In the same manner,

$QA=\frac{y}{\sqrt{3}}\ \textnormal {and} \ BR=\frac{z}{\sqrt{3}}$

Using the cosine rule

$(PQ)^2=(\frac{x}{\sqrt{3}})^2+(\frac{y}{\sqrt{3}})^2-2\frac{x}{\sqrt{3}}\frac{y}{\sqrt{3}}cos(60+\theta)$

$(PQ)^2=\frac{x^2}{3}+\frac{y^2}{3}-\frac{2xy}{3}cos(60+\theta)$

$\textnormal{Use a trigonometric identity to expand}\ cos(60+\theta)$

$(PQ)^2=\frac{x^2}{3}+\frac{y^2}{3}-\frac{2xy}{3}(cos60cos\theta-sin60sin\theta)$

$(PQ)^2=\frac{x^2}{3}+\frac{y^2}{3}-\frac{2xy}{3}(\frac{1}{2}cos\theta-\frac{\sqrt{3}}{2}sin\theta)$

Now from the original triangle ABC

$cos\theta=\frac{x^2+y^2-z^2}{2xy}\ \textnormal {and}\ A=\frac{1}{2}xysin\theta\ \textnormal {where} A\ \textnormal {is the area.}$

$\textnormal{Substitute into}\ (PQ)^2$

$(PQ)^2=\frac{x^2}{3}+\frac{y^2}{3}-\frac{2xy}{3}(\frac{x^2+y^2-z^2}{4xy}-\frac{\sqrt{3}A}{xy})$

Multiply both sides by 3 and expand and simplify

$3(PQ)^2=x^2+y^2-2xy(\frac{x^2+y^2-z^2}{4xy}-\frac{\sqrt{3}A}{xy})$

$3(PQ)^2=x^2+y^2-(\frac{x^2}{2}+\frac{y^2}{2}-\frac{z^2}{2}-2\sqrt{3}A)$

$3(PQ)^2=\frac{x^2}{2}+\frac{y^2}{2}+\frac{z^2}{2}+2\sqrt{3}A$

We are going to do the same for QR

$(QR)^2=\frac{y^2}{3}+\frac{z^2}{3}-\frac{2yz}{3}(cos(60+\alpha)$

$(QR)^2=\frac{y^2}{3}+\frac{z^2}{3}-\frac{2xy}{3}(cos60cos\alpha-sin60sin\alpha)$

$(QR)^2=\frac{y^2}{3}+\frac{z^2}{3}-\frac{2xy}{3}(\frac{1}{2}cos\alpha-\frac{\sqrt{3}}{2}sin\alpha)$

Once again, from triangle ABC

$cos\alpha=\frac{y^2+z^2-x^2}{2yz}\ \textnormal {and}\ A=\frac{1}{2}zysin\alpha\ \textnormal {where} A\ \textnormal {is the area.}$

$\textnormal{Substitute into}\ (QR)^2$

$(QR)^2=\frac{y^2}{3}+\frac{z^2}{3}-\frac{2yz}{3}(\frac{y^2+z^2-x^2}{4yz}-\frac{\sqrt{3}A}{yz})$

Multiply both sides by three and expand and simplify

$3(QR)^2=y^2+z^2-2yz(\frac{y^2+z^2-x^2}{4yz}-\frac{\sqrt{3}A}{yz})$

$3(QR)^2=y^2+z^2-(\frac{y^2}{2}+\frac{z^2}{2}-\frac{x^2}{2}-2\sqrt{3}A)$

$3(QR)^2=\frac{x^2}{2}+\frac{y^2}{2}+\frac{z^2}{2}+2\sqrt{3}A$

We can do the same for PR

$(PR)^2=\frac{x^2}{3}+\frac{z^2}{3}-\frac{2xz}{3}cos(60+\gamma)$

$(PR)^2=\frac{x^2}{3}+\frac{z^2}{3}-\frac{2xz}{3}(cos60cos\gamma-sin60sin\gamma)$

$(PR)^2=\frac{x^2}{3}+\frac{z^2}{3}-\frac{2xz}{3}(\frac{1}{2}cos\gamma-\frac{\sqrt{3}}{2}sin\gamma)$

And from triangle ABC

$cos\gamma=\frac{x^2+z^2-y^2}{2xz}\ \textnormal{and}\ A=\frac{1}{2}sin\gamma\ \textnormal{where A is the area.}$

$\textnormal{substitute into}\ (PR)^2$

$(PR)^2=\frac{x^2}{3}+\frac{y^2}{3}-\frac{2xz}{3}(\frac{x^2+z^2-y^2}{4xz}-\frac{\sqrt{3}A}{xz})$

Multiply both sides by three and expand and simplify

$3(PR)^2=\frac{x^2}{2}+\frac{y^2}{2}+\frac{z^2}{2}+2\sqrt{3}A$

Hence, PQ=PR=RQ and triangle PQR is equilateral.

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Filed under Interesting Mathematics, Napoleon's Triangle

Category Archives: Interesting Mathematics

Derangements

Interesting Sum

Volume and Surface Area of a Conical Frustrum

Interesting Equation

Infinite Product Expansion of cos (x)

Fibonacci Sequence – Finding the Closed Form

Napoleon’s Theorem

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