Blogs, Grading, Mathematical Reasoning, Quizzes

An impossible quiz

I have often seen quizzes on Facebook or other social media stating that they are extremely difficult and only a genius could answer, say, 5 out of 10. They appear about different topics and have questions pertaining to many different areas. Therefore, I wanted to write my own quiz. These questions all pertain to STEM in some shape or form. If you can even get 1 out of 10, then you are truly a genius.

Integer Problems

  1. Find two integers, \(p,q\) such that \(\frac{p}{q}^{3}=2\).
  2. Find a natural number \(n\) such that this \(n\) can be factored in two ways into the product of prime natural numbers (a reordering of terms counts as the same factorization ex \(2*2*3=2*3*2=2^{2}*2\) are all considered equal).
  3. Find an integer, \(k\), such that \(k^{2} \equiv 2 \text{mod} 4\).  (See Modular Arithmetic for definition of \(\equiv\) mod \(n\)).
  4. Find a finite set of integers, \(A\), such that \(A\) does not have a smallest or a largest element (where smallest and largest is under the usual ordering of integers).

Coloring Problems

  1. Draw a map (where each region is bounded and connected) such that, if you were to color the map in such a way that no regions sharing a boundary (larger than a single point) can be colored the same color, that 5 colors would be required.
  2. For the graph below, labeled Graph 1, color the edges with 4 colors in such a way that any two edges incident at a point cannot share a common colorgraph1
    Graph 1

For the following you are working within the Euclidean plane.

  1. Suppose that you have a cube with a given side length.  Using only a straight-edge and compass, find the exact side length of a cube with twice the volume.
  2. Suppose that you have a circle with a given radius.  Using only straight-edge and compass, use the radius to draw a square with the same area as the circle.
  3. Show that for any two given lengths \(a\) and \(b\), construct a length \(c\) such that \(c\) divides both \(a\) and \(b\).  That is, \(nc=a\) and \(mc=b\) for some natural numbers \(n\) and \(m\).


  1. Suppose that you are the person in the picture below.  Your city is broken into 4 pieces each surrounded by rivers.  You aren’t able to swim, jump over, or walk through any of the rivers.  However, there are bridges over the rivers connecting the pieces of the city as drawn below.  Find a path that will allow you to walk over each bridge exactly once and finish where you started.  For the picture land is gray, water blue and the bridges are the .

Extra Help

If you would like to learn more about these problems, I have written about most of them in other posts.  Below, I have provided links to the posts so that you can learn more about the problem.

Integer Problems

  1. We look at this problem in That’s irrational.
  2. This problem appears in Fundamental Theorem of Arithmetic.
  3. See Modular Arithmetic for definition of \(\equiv\) mod \(n\).  Then over the set of \(\left\{0,1,2,3\right\}\) find what each one of these squared is mod4.
  4. See Lemma .8 in the write up for That’s irrational.

Coloring Problems

  1. This problem appears in Coloring Game!
  2. A similar problem appears in More Coloring Games.


  1. This problems appears in Double the Size of that Altar.
  2. This is referenced What is most interesting: question, answer, or explanation?
  3. See Comparing Fractions with Pythagoras for examples of such problems.


  1. This is referenced in What is most interesting: question, answer, or explanation?


As a reward for reading this far, I will attempt to save you some time toiling over these problems.  Everyone of them is impossible.  We in fact have shown that most of them are impossible in the mentioned posts.  The two that we haven’t proven on this blog yet are the problems of squaring a circle and the bridges of Konigsberg, which we may get to at some point.  While this may not be satisfying, the point I take from the example is one I try to remember when writing an exam.  It is always possible to make an exam no one will pass, therefore, it should be my goal to make one that correctly identifies what a person knows and what a person does not.  In making a test too difficult, or too easy, the knowledge of the person taking it gets lost, so it is important to make a well made exam in order to provide proper feedback to students.

On the other hand, while the questions may not be solvable, there has been a wealth of knowledge obtained from attempting to answer these questions.  We have seen entire fields of study be born out of these problems, so attempting the impossible is not a futile endeavor.  I hope you enjoyed the questions and learned something from the linked posts.   If you enjoyed this, or any of the other posts, please let me know by using the like button, and by sharing them on Social Media.

2 thoughts on “An impossible quiz”

  1. Thank you to everyone that read this post. It seemed like there was a lot of interest in the problems. If you would like to look over some questions that are possible, please visit

    Forums/General/Problem of the Week

    Check back regularly as we will update these problems weekly.

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