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Tricky and Unusual Math Problems

April 26, 2012

As if mathematics wasn't difficult enough, students are faced with tricky exam questions. Students of my generation preferred multiple choice questions over other types. Our multiple choice questions were posed as follows:

Which of one of the following is a prime number?
a) 4
b) 9
c) 11
d) 21
e) 49

Students of my children's generation faced the reinvented version of the multiple choice question of the type,

Mary has three coins in her purse.[1] These coins can have a combined value of
a) 3, 7 or 12, but not 14
b) 75 or 40, but not 8
c) 7, 12 or 52
d) All but one of the above
e) All of the above

Problems like this are a sneaky way of putting three problems into the space allotted for one.

There are many jokes that are posed as unanswerable questions, such as, "A man walked into a restaurant. What color socks was he wearing?" There's actually a math problem that sounds just as unanswerable, but it does have an answer.

This problem is called, variously, "The Census-Taker Problem" and "The Ages of Three Children Problem." I was reminded of this problem by a recent posting on arXiv by I.J.L. Garces and Mark L. Loyola of the Department of Mathematics of the Ateneo de Manila University.[2] Here's their version of the problem.
A census taker knocks on a door. A mother answers.
The census taker says, "I need to know the number of children you have, and their ages."
The woman responds in puzzle-ese, "I have three daughters, the product of their ages is 36, and the sum of their ages is equal to the house number next door."
The census taker, who never wastes questions, computes for a while and then asks, "Does your oldest daughter love dogs?"
The mother answers affirmatively.
The census taker says, "Thank you. I now know the ages."
What are the ages of the children?

As if this isn't confusing enough, what's that question about dogs?

The simple answer is that there are just two sets of three numbers that satisfy the condition that the product is 36; namely, {9,2,2} and {6,6,1}. The question about the dog is to determine if there is an older daughter, resulting in the {9,2,2} combination. There is a chance that the mother might have said that none of her children like dogs, so another question would be needed; but then the problem wouldn't be so memorable.

Another paper, posted by Tanya Khovanova on arXiv, presents many tricky arithmetic problems.[3] Khovanova writes an interesting blog[4] and she has a web site, also.[5] Here are a few of these tricky arithmetic problems.[3]

A stick has two ends. If you cut off one end, how many ends will the stick have left?
A square has four corners. If we cut one corner off, how many corners will the remaining figure have?
3.Peter had ten cows. All but nine died. How many cows are left?
A patient needs to get three shots. There is a break of 30 minutes between shots. Assuming the shots themselves are instantaneous, how much time will the procedure take?
You are running a race and you pass the person who was running second. What place are you now?
One hundred percent of the fish in a pond are goldfish. I take 10% of the goldfish out of the pond. What percentage of goldfish does the pond now contain?
The Davidsons have five sons. Each son has one sister. How many children are there in the family?

Martin Gardner

Tanya Khovanova presented her tricky arithmetic problems at the 2012 Gathering for Gardner

Martin Gardner popularized mathematics through his Mathematical Games column in Scientific American.

(Photo by Konrad Jacobs, via Wikimedia Commons))


  1. US coins have the following cent values: 1, 5, 10. 25, 50, 100.
  2. I.J.L. Garces and M.L. Loyola, "Revisiting a Number-Theoretic Puzzle: The Census-Taker Problem," arXiv Preprint Server, April 10, 2012.
  3. Tanya Khovanova, "Tricky Arithmetic," arXiv Preprint Server, April 13, 2012.
  4. Tanya Khovanova's Math Blog.                                   
  5. Tanya Khovanova Web Site.                                   

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Linked Keywords: Mathematics; student; exam; baby boomer; multiple choice; prime number; Generation Y; joke; The Census-Taker Problem; The Ages of Three Children Problem; arXiv; Department of Mathematics; Ateneo de Manila University; dog; set; product; Tanya Khovanova; arithmetic; Gathering for Gardner; Martin Gardner; Scientific American; Konrad Jacobs; Wikimedia Commons.