Rong But Reasonable: Chapter 8
MCQs or Mildly Confusing Questions
Welcome to the eighth post in the series Rong But Reasonable. In these articles we examine the ‘reasonableness’ of wrong answers to math questions. Our aim is to explore the common mathematical misconceptions among students. In addition, we also reflect upon the fallacies educators and others fall victim to. Previous articles in the series can be found here.
In this week’s post I rant about two kinds of MCQs commonly encountered in exams. This format is usual in primary grades. I have seen similar questions in various competitive exams such as JEE or UPSC as well. What is this question type so universally regarded as a tool to measure intelligence that rankles me enough to write a post bashing it?
Bugbear 1: What comes next?
1, 2, 4, 8, 16, ?
(A) 20 (B) 32 C) 77 (D) -10.
Most people intuitively answer 32. Why? The numbers get doubled successively. Another way to describe is that it is a geometric progression. Seems obvious and harmless.
What if I told you the answer is 77. The sequence is
1, 2, 4, 8, 16, 77, 145, 668, …
How? We apply the following rules:
a) Reverse the digits in the number
b) Add it to the original number
c) Sort the digits in the total in ascending order
We can see how this procedure generates the sequence.
8 + 8 = 16. 16 + 61 = 77. 77 + 77 = 154, upon sorting gives 145. 145 + 541 = 686, upon sorting gives 668. (I found this amusing sequence in Ian Stewart’s Casebook of Mathematical Mysteries)
Or what if I told you that the answer to the original question is 20. The sequence is
1, 2, 4, 8, 16, 20, -9, -132, … where the nth term is given by the polynomial
Or what if I told you the answer is -10, with the nth term given by
In fact, the answer could be anything.
Given any sequence of numbers, it is possible to find a polynomial which fits the sequence. This polynomial can be found using a method called Lagrange interpolation.
We can view it as a precursor to Fourier transform. A brief description of Lagrange interpolation is given at the end of the article.
(Incidentally, my masters’ research was based on this topic, although in a different context)
The point I wish to highlight is that we can give any answer to the question ‘what comes next in the sequence’ and give a sufficiently obfuscated justification.
Bugbear 2: Odd one out
A couple of years ago, a ten-year-old girl posed me this question.
Who is the odd one out?
(A) Iron-Man (B) Spiderman C) Captain Marvel (D) Batman.
Can you find the answer? What is your reason?
Now, I consider myself a sophisticated person. I have extraordinary reasoning abilities. I am also sufficiently educated about pop-culture, at least enough to sustain a conversation with a zoomer. I could answer the question instantly.
‘Of course, it’s Batman’, I said, ‘All of the others are Avengers, part of MCU. Batman is part of DC Comics.’
‘Wrong!’ she said, ‘It’s Captain Marvel. She is a woman. Others are men.’
I felt my answer was better, because I could give one more reason why Batman was the odd one. ‘Batman has no superpowers. All the others have.’ To this she countered, ‘That’s not true. Iron-Man has no superpowers either. In fact, I can also give one more reason why Captain Marvel is the odd one out. Others wear a mask, but she doesn’t.’ Touche!
This got me thinking. If she were my teacher and gave me this question in an exam, she would have given me a zero for my answer. If I were the teacher and gave the same question to her in an exam, I would have given her a zero for her answer.
Here is another question. Take a second to answer this along with the reason.
Find the odd one out among these: (A) frog (B) deer C) crow (D) leopard.
What if I told you that your answer is correct irrespective of your choice?
Take a minute to think why each one of the animals could be the odd one out.
- The answer is ‘crow’. It has wings, others don’t. It has a beak, others don’t. It can fly, others can’t. I could also argue that a crow does not have spots, but others do. This is partially true. Some species of frogs and deer, and all leopards, have spots. But there are also frogs which do not have spots.
- The answer is ‘frog’, because it lives underwater, and reproduces through external fertilization. Other animals in the list do not exhibit these characteristics. I could also argue that there is more species diversity among frogs than the other animals, i.e. there are more kinds of frogs than deer, crows or leopards.
- The answer is ‘deer’, because that is the only primary consumer in the list. All the other animals are secondary or tertiary consumers in the food chain. Also, this is the only animal that does not live on trees. I could also argue that DEER is that only animal that makes a sensible word when written backwards (REED). It is also the only word where a letter is repeated within the word.
- The answer is ‘leopard’ because that is the only word not four letters long. It is the only word with more than one vowel.
(Unfortunately, my expertise in biology is limited, hence I need to resort to verbal gymnastics to justify why leopard is the odd one out!)
I could also argue that ‘crow’ is the odd one out because it lays eggs. This is not correct since a frog also lays eggs. Crow might very well be the odd one out: we have already provided a few reasons why. But ‘laying eggs’ is not one of them. This is an example of getting the correct answer through a wrong method or reason.
I could argue that ‘leopard’ is the odd one out because it has a tail. This is also incorrect because crows and deers also have a tail. A frog too has a tail, when it is a tadpole. So, this cannot be a valid reason.
As a fun exercise, you can find features that are common to a pair of animals in the list. For example, both crow and leopard have claws, but others don’t. Both deer and leopard are mammals, etc. Can you find a justification of every pair of animals?
We have seen that it is possible to justify our answers in different ways. In addition to having different answers, we have different kinds of answers. We also have different levels of reasonableness.
We have justifications based on biology. We have justifications based on language. We have justifications that are only partially true. These questions provide an excellent opportunity to provoke reasoning and lateral thinking. But…
These questions are treated as objective-type, which means they have a single right/wrong answer. Such questions are easy to answer and easy to evaluate in an exam. This perspective loses so much of the nuances these questions offer. ‘What comes next’ and ‘odd one out’ are even used in high-stakes exams like JEE or UPSC or bank entrance exams. The tragedy is that these questions do not even serve their primary purpose to test aptitude, knowledge or reasoning. We have seen in the previous examples as to how ‘odd one out’ questions can be used to test for knowledge. Unfortunately, these questions now simply test the mind-reading abilities of the students. Among the many patterns that could be discovered, the student must write the one specific pattern or rule that the examiner has in mind. The examiner will reward you if you think like them and find it.
To summarize, ‘what comes next’ and ‘odd one out’ are convenient tools for assessment. However, they have far greater potential to sharpen the thinking skills of students. Let us use these questions to spark originality among students instead of using them to check if the students can guess what is on our minds.
Extras:
Here is a talk by Alex Bellos, a well-known math popularizer. In this entertaining talk, he discusses a classic odd one out question.
Lagrange interpolation: If a_1, a_2, …, a_n is a sequence, then there exists a polynomial f(x) of degree n such that f(1) = a_1, f(2) = a_2, …, f(n) = a_n. The following example will illustrate the method to find such a polynomial.
L_0 will take the value 1 at x = 0, 0 at x = 1 and x = 2. Similarly, L_1 and L_2 will take the value 1 at x = 1 and x = 2 respectively. The polynomial we seek is just the appropriate linear combination of these three polynomials. Simplifying, we get
