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## Monday, 22 September 2014

### Sunday Afternoon Maths XXIX Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

#### Three Squares

Draw three more squares and add these lines (I have coloured the angles to make equal angles clearer):
Triangles $$ACE$$, $$LDK$$ and $$IKE$$ are congruent, so angle $$KDL$$ is equal to $$\beta$$.
The congruence of these triangles tells us that angles $$DKL$$ and $$EKI$$ add up to a right angle, so angle $$EKD$$ is also a right angle.
The congruence of the triangles also tells us that $$KD$$ and $$KE$$ are the same length and so angle $$EDK$$ is the angle in an isosceles right-angled triangle. $$\alpha$$ is also the angle in an isosceles right-angled triangle, so these two angles are equal.
Therefore $$\alpha+\beta+\gamma=90^\circ$$.
##### Extension
The diagram shows three squares with diagonals drawn on and three angles labelled.
What is the value of $$\alpha+\beta+\gamma$$?

#### Equal Opportunity

Let $$p_1$$, $$p_2$$, ..., $$p_6$$ be the probabilities of getting 1 to 6 on one die and $$q_1$$, ..., $$q_6$$ on the other. The probability of getting a total of 2 is $$p_1q_1$$ and the probabilty of getting a total of 12 is $$p_6q_6$$. Therefore $$p_1q_1=p_6q_6$$.
If $$p_1\geq p_6$$ then $$q_1\leq q_6$$ (and vice-versa) as otherwise the above equality could not hole. Therefore:
$$(p_1-p_6)(q_1-q_6)\leq 0$$ $$p_1q_1-p_6q_1-p_1q_6+p_6q_6\leq 0$$ $$p_1q_1+q_6p_6\leq p_1q_6+p_6q_1$$
The probability of rolling a total of 7 is $$p_1q_6+p_2q_5+...+p_6q_1$$. This is larger than $$p_1q_6+p_6q_1$$, which is larger than (or equal to) $$p_1q_1+q_6p_6$$, which is larger than $$p_1q_1$$.
Therefore the probability of rolling a 7 is larger than the probability of rolling a two, so it is not possible.
##### Extension
Can two $$n$$-sided dice be weighted so that the probability of each of the numbers 2, 3, …, 2$$n$$ is the same?
Can a $$n$$-sided die and a $$m$$-sided die be weighted so that the probability of each of the numbers 2, 3, …, $$n+m$$ is the same?

#### Double Derivative

(i) $$\frac{dy}{dx}=1$$, so $$\frac{d}{dy}\left(\frac{dy}{dx}\right)=0$$
(ii) Differentiating $$y=x^2$$ with respect to $$x$$ $$\frac{dy}{dx}=2x$$. Let $$g=\frac{dy}{dx}$$. By the chain rule:
$$\frac{dg}{dy}=\frac{dg}{dx}\frac{dx}{dy}$$ $$=2\frac{1}{2x}$$ $$=\frac{1}{x}$$
So $$\frac{d}{dy}\left(\frac{dy}{dx}\right)=\frac{1}{x}$$
(iii) By the same method, $$\frac{d}{dy}\left(\frac{dy}{dx}\right)=\frac{2}{x}$$
(iv) $$\frac{d}{dy}\left(\frac{dy}{dx}\right)=\frac{n-1}{x}$$
(v) $$\frac{d}{dy}\left(\frac{dy}{dx}\right)=1$$
(vi) $$\frac{d}{dy}\left(\frac{dy}{dx}\right)=-\tan(x)$$
##### Extension
What is
$$\frac{d}{dy}\left(\frac{dy}{dx}\right)$$
when $$y=f(x)$$?