Sweet birthdays in math exams!
R. Ramanujam, The Institute of Mathematical Sciences, Chennai
Hannah's Sweets
These are exam times and we keep hearing children exclaiming, "Oh that was easy!", or "What a weird question paper!", or "Whatever did they mean in that question?", and so on.
Here is a question that stirred up people greatly a few years ago. It was a problem asked in the GCSE mathematics examination in June 2015 (equivalent of our Class X in the UK):
Hannah has a bag containing a total of n sweets, of which 6 are orange. If the chances of Hannah picking two orange sweets one after the other is one-third, use this to prove that n^2-n-90 = 0.
How does one reason with this? Many children simply see this as `factorization', or `solving the quadratic equation' and are hence puzzled by all the fuss about Hannah and her sweets. It would be so much easier if it was simply a matter of finding the value of n. And apparently that is what most children did, missing the question asked by the examiners.
The information given is about probability. What is the probability of the first sweet Hannah picked being orange? That's easy: there are 6 orange out of n sweets, the probability of picking an orange sweet is 6/n. Well then, what is the probability that the second sweet Hannah picked was orange as well? Now there were (n-1) sweets left over, with 5 among them being orange, so we get 5/(n-1). Now what is the probability of these two events happening, one after the other? Since the two events are independent of each other, it is the product of the two probabilities:
      (6/n) * (5/(n-1)).
But this is given to be one-third. So we get the equation:
      (6/n) * (5/(n-1)) = (1/3).
Simplifying, we have
      (30/(n^2 - n)) = (1/3).
Hence
      90 = n^2 - n,
and we get the given equation.
If you really want to solve this equation, you will get n=19. Substitute and check if you don't know how to solve a quadratic equation.
Cheryl's Birthday
Before Hannah there was Cheryl. A few months earlier, in April 2015, an exam question from Singapore caused a great deal of discussion all over the world. This was an exercise in pure logic.
Albert and Bernard friends are new friends of Cheryl, and they want 
to know when her birthday is. Cheryl gives them a list of 10 possible dates:
in May: 15, 16 or 19;
in June: 17 or 18;
in July: 14 or 16;
in August: 14, 15 or 17.
Cheryl then tells Albert separately the month of her birthday and and Bernard separately the day of her birthday. 
The following conversation takes place between Albert and Bernard.
Albert: I don't know when Cheryl's birthday is, but I know that you don't know it either.
Bernard: Before you said this, I didn't know Cheryl's birthday, but I know it now!
Albert: Then I also know Cheryl's birthday!
So when is Cheryl's birthday?"
Such questions require you to think calmly about the given information and reason out the possibilities.
Albert is told the month. In each of them there is more than one possibility, so of course he does not know the day.
Bernard is told the day. If it is 14, 15, 16 or 17, he would be uncertain, but if it is 18 then he immediately knows that it is "June 18". Similarly if he were told 19, it has to be "May 19".
Albert says that he knows that Bernard does not know the birthday. If he had been told May or June, there was no way for him to rule out 18 or 19 and hence he could not be sure of Bernard's ignorance. Hence we conclude that Albert was told either July or August. But so can Bernard conclude this as well!
Notice that Bernard says that he knows the birthday now. Clearly, Bernard heard 14, 15, 16 or 17. Had he heard 14, he would still be unsure whether it was July or August, so he must have heard 15, 16, or 17. So we conclude that the birthday is August 15, July 16, or August 17. But then so can Albert conclude this as well.
But then Albert says he knows the birthday too, so he could not have heard August. (If so he would be unsure whether it was the 15th or the 17th.) Hence Cheryl's birthday must be on July 16.
This line of problems follows the famous "sum--product" puzzles popularized by Martin Gardner, the great American puzzle-maker (1914 to 2010). Jantar Mantar has carried articles on him and on such puzzles. Children in Singapore would have benefited from reading Jantar Mantar for cracking their Board examinations!