Sunday Afternoon Maths XVII Answers & Extensions

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

Maths Jam

15^th February 2022.

Extension

What is the latest date in the month on which Maths Jam can fall and when will this next happen?

N

(b) implies that the digits of \(N\) are all 1 or 7, so \(N\) can only be 111, 117, 171, 177, 711, 717, 771 or 777. These are all divisible by 3, so no such integers \(N\) exist.

Extension

Consider 21-digit integers \(N\) such that:

(a) \(N\) is not exactly divisible by 2, 3 or 5.

(b) No digit of \(N\) is exactly divisible by 2, 3 or 5.

How many such integers \(N\) are there?

Square Numbers

Let \(a^2\) and \(b^2\) be the two square numbers.

$$2(a^2 +b^2 ) = 2a^2 +2b^2$$ $$= a^2 + 2ab + b^2 + a^2 - 2ab + b^2$$ $$= (a+b)^2 +(a-b)^2$$

Extension

Source: Lewis Carroll's Games & Puzzles

Prove that 3 times the sum of 3 squares is also the sum of 4 squares.

Differentiate This

$$f(x)=e^{x^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}} }$$ $$=e^{e^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}\ln{x}} }$$ $$=e^{e^{ \ln{\left(\ln{x}\right)}} }$$ $$=e^{\ln{x} }$$ $$=x$$

Therefore:

$$f'(x)=1$$