Mathematical Quiz: 1

This post is just a first attempt at creating a mathematical quiz. I'm still thinking about the best way to present such a quiz from the wide variety of online resources available. The target audience is an important consideration. The first seven questions of this particular quiz is accessible to those you have completed a course in high school mathematics. The last three questions however, would not be but would serve to stimulate interest and get them to follow the suggested links. This whole quiz concept is a work in progress so I'll keep experimenting with quiz content and design.

Here is a set of ten mathematical questions that will test your understanding of Mathematics and perhaps help you to learn things of interest in the process. You should not use a calculator (except for Question 6) or reference material to answer these questions. Just rely on your own resources.

Questions:

1. Evaluate \(2^{3^2}\)
   
   
2. \(\pi\) represents the ratio of a circle's diameter to its circumference while \(e\) is the base of the natural logarithms. What is the product of these two numbers?
   
   
3. Evaluate \( \dfrac{1}{0!}\)
   
   
4. Evaluate \(4 + 8 \div 4 \times 2\)
   
   
5. Will \(2100\) be a leap year?
   
   
6. In a random group of people, how many are needed so that the probability of two people sharing the same birthday is about 50%? You can use a calculator for this problem.
   
   
7. Can you find the smallest integer that can be written as \(x^2+xy+y^2 \) in two different ways with \(x \geq 0\) and \(y \geq 0\)? Hint: it's smaller than 50.
   
   
8. A happy number is one that reduces to 1 with repeated sums of squares of digits. For example, \(13 \rightarrow 1^2+3^2 = 10 \rightarrow 1^2+0^2 = 1\). What happens to numbers that aren't happy?
   
   
9. \(5=2^2+1^2\) but \(7\) can't be written as a sum of two squares. Using this information, try to decide whether the prime number \(1009\) can or cannot be written as a sum of two squares. Hint: use modular arithmetic. 
   
   
10. Who is this German mathematician depicted below? Hint: his first name is Georg. He was born in 1845 and died in 1918.

Answers:

1.  The rule is that the calculation proceeds from the top downwards and so we calculate \(3^2=9 \) first, then \(2^9=512\). Proceeding from the bottom up, we would evaluate \(2^3=8\) and then \(8^2=64\) but this is incorrect. Thus the answer is 512.
   
   
   Comment: I've written about this in a blog post titled Power Towers and Tetration. This is a simple but important principle to understand and is a sort of extension of the BOMDAS rule (Brackets, Of, Multiplication, Division, Addition, Subtraction).
   
   
2. This is definitely a trick question. The answer is \(pie\).
   
   Comment: there's always room for humour in mathematics, provided it's not overdone. 
   
   
3. It needs to be remembered that \(0!=1\) and thus the answer is 1.
   
   Comment: many former high school students would remember that zero factorial is 1 so this is not as difficult as it looks.
   
   
4. To prevent mistakes put a bracket around division and multiplication before proceeding from left to right. This gives:
   
   \(4 + ((8 \div 4 )\times 2)=4 + (2 \times 2)=4+4=8\)
   
   Comment: this will trick a lot of people but it's still an elementary problem that even an upper level primary student should be able to handle.
   
   
5. End of century years must be divisible by \(4\) and \(100\). While \(2100\) is divisible by \(100\), it is not divisible by \(4\) and thus it is not a leap year.
   
   Comment: this is not widely known but it should be and so this problem will inform those who weren't familiar with the rule.
   
   
6. This is the famous birthday problem and the answer is 23 people. I've written about this is a blog post titled 23.
   
   
   Comment: the number is somewhat counter-intuitive in that it's much smaller than one might expect. It's an interesting problem that doesn't require any high level mathematics but will require a calculator (hence the exemption).
   
   
   Here is a brief explanation taken from my previously mentioned blog post:
   • With 23 people we have 253 pairs: \(\dfrac{23 \times 22}{2}=253\)
   • The chance of two people having different birthdays is \(1−\dfrac{1}{365}=\dfrac{364}{365}=0.997260\)
   • Makes sense, right? When comparing one person's birthday to another, in 364 out of 365 scenarios they won't match. Fine. But making 253 comparisons and having them all be different is like getting heads 253 times in a row - you had to dodge "tails" each time. Let's get an approximate solution by pretending birthday comparisons are like coin flips.We use exponents to find the probability:
   • \( \left (\dfrac{364}{365} \right )^{253}=0.4995 \approx 50 \%\)
   • Our chance of getting a single miss is pretty high (99.7260%), but when you take that chance hundreds of times, the odds of keeping up that streak drop. Fast.
   
   
7. The smallest integer is \(49=0^2+0 \times 7+7^2=3^2+3 \times 5+5^2\). Such numbers are called Loeschian numbers and I've written about them in this post.
   
   Comment: this is easy to work out with a little trial and error.
   
   
8. It shouldn't take too long for someone to realise that numbers that aren't happy end up in the loop {4,16,37,58,89,145,42,20}. I've written about these in a post titled Happy Numbers.
   
   Comment: the discovery takes just a little trial and error.
   
   
9. All primes of the form \(4k+1\) where \(k \geq 1\) can be written as a sum of two squares. Now \(1009 \div 4\) leaves a remainder of \(1\) so it is of the form \(4k+1\) and can be written as a sum of two squares (\(15^2+28^2\)). I've written about these in a post titled Sum of Two Squares. 
   
   
   Comment: this is a little difficult but the hint to use modular arithmetic should nudge people in the right direction.
   
   
10. His name is Georg Cantor and he is the "father" of set theory. You can read more about him by following this link.
    
    
    Comment: the first name is "Georg" and other hints will eliminate the well-known mathematics so some people may guess this because "Cantor" is reasonably well-known.

Since creating this quiz I've modified and improved the questions in various ways, so it's been a useful exercise. I still have to decide on the best way to present them. I may experiment with various formats and report back on this post as I'll use this quiz as the content.

ADDENDUM:

I've made use of QUIZIZZ to create a multiple choice quiz using 9 out of the 10 questions. Question 2 wasn't suitable for multiple choice so I've replaced it with another one involving identification of primes. A negative is that the site requires the setting up of a class and the addition of the quiz to that class as homework. Anyone wanting to take the test needs to set up an account by visiting https://quizizz.com/join/class and then use the class code which is M214707.

There are other negatives. As far as I can tell there is no support for LaTeX and so any mathematical expressions have to be included as images. However, the images are easily imported and display well so it's not a major issue. Any revisions mean that the image must be deleted and a new one imported.