Understandable questions which are hard for non-mathematicians but easy for mathematicians

Question

A friend of mine has set me the challenge of finding an example of the following:

Is there a question, that everyone (both mathematicians and non-mathematicians) can understand, that most mathematicians would answer correctly, instantly, but that most non-mathematician would struggle to solve/take longer to arrive at a solution?

When I say mathematician, I mean a person who has studied maths at degree level. The key feature of such a question is that it should be understandable to an average person. I realise, of course, that this is a subjective question - what does 'most mathematicians' mean, what would 'most mathematicians' be able to answer? Nonetheless, I would be interested to hear peoples' opinions and ideas:

Can you think of a question, that in your opinion, is an example of the above?

---

An example of such a question, I think, would provide a good way of explaining to people how mathematicians think. It could also be a good teaching tool (i.e to show how mathematicians approach problem solving).

My friend suggested the following question:

Does there exist a completed Sudoku grid with top row $1,2,3,4,5,6,7,8,9$?

I won't give the answer (so that you can see for yourself if it works!). When we asked this to fellow maths researchers, nearly all were able to give the correct answer immediately. When I asked the undergraduates that I teach, most of them (but not all) could answer correctly and pretty quickly. I like this question but I'm sure there's a better one.

Thanks!

---

Edit: As for the Sudoku question, most of the researchers I asked answered in 10 seconds with the solution:

Yes. You can relabel any Sudoku (i.e swaps sets of numbers - change all $1$s for $2$s for example) and still have a valid Sudoku solution. So, in a sense all Sudoku are equivalent to a Sudoku with one to nine in the first row.

Answer 1

A good example is the Bridges of Königsberg puzzle. An important city in 18^th century Prussia was the city of Königsberg (modern day Kaliningrad, a Russian enclave) which had seven bridges. The residents played a game: try to cross every bridge precisely once. No one could solve this puzzle for a long time.

Euler proved this was impossible. Every mathematician knows the solution and how to solve similar problems, although this is by training rather than their own mental guile :-)

Answer 2

Here's one: is $111^3-58^3$ a prime number? Most non-mathematicians would compute that number, get $1\,172\,519$ and then waste some time (or perhaps a lot of time) in search of prime factors.

A mathematician would say that that number is equal to $(111-58)\times(111^2+111\times58+58^2)$ and therefore it can not possibly be a prime number.

Answer 3

Here's a famous one:

Cover, if you can, a chess $8\times 8$ with $31$ pieces $2\times 1$ leaving empty the upper-right and bottom-left corners.

I saw many people wasting several hours on possible coverings. On the other hand, a mathematician (or a good high school student) would solve it quite quickly (which is: a piece $2\times 1$ cover a black and a white small square; but the upper-right and bottom-left corners have the same colour).

Answer 4

Here's one I like. You have an electronic scale, and ten piles of ten coins each. Exactly one pile out of the ten is made of fake coins; the other nine piles are made of regular coins. A regular coin weighs 10 grams; a fake one weighs only 9 grams.

You are allowed to take any number of coins and weigh them once. How do you find out which pile is made of fake coins?

Answer 5

The 14-15 puzzle was introduced by Sam Lloyd* in the early 20th Century. It was a 4x4 sliding-square puzzle, like the picture puzzles you find in museums. The squares were numbered: $$ \begin{array}{cccc} 1&2&3&4\\ 5&6&7&8\\ 9&10&11&12\\ 13&15&14& \end{array} $$ (The last square is empty so that you can slide the squares around.)

Challenge. Swap the 14 and the 15, so that all the squares end up in the correct order.

The solution is quite elegant, if you know enough maths:

Solution. This is impossible. Swapping two squares corresponds to an "odd" permutation. However, every permutation which takes the blank square back to itself is an "even" permutation (why?).

A related, and more modern, puzzle would be: can you rotate a single square in a Rubik's cube? The answer is no, for the same reason as above.

*A specific case of a more general puzzle, which he had nothing to do with but claimed credit for. See wiki for more details...

Answer 6

This problem is described in Professor Stewart's Hoard of Mathematical Treasures as "a time-honoured way to make money in a pub, requiring three cups and one mug. (The mug is human, and should be moderately intoxicated for greater gullibility.)"

You have three cups. Two of them are inverted and one is upright. You can only flip two cups at a time. Your objective is to flip all the cups down.

This is impossible because of a parity argument (the number of inverted cups stays even). This same argument shows that the problem starting from $n$ upright cups, where each move inverts $m$ cups, is impossible iff $n$ is odd and $m$ is even.

Answer 7

If most mathematicians shall be able to solve it instantly and most non-mathematicians shall be able to solve it but use longer time, I would go for something real simple like:

What is the sum of all numbers from 1 to 100

The more the upper limit is increased (e.g. 100000 instead of 100), the more difficult it will be for non-mathematicians while the upper limit wouldn't make much difference for mathematicians.

Answer 8

All existing answers are about math problems. Studying mathematics, however, improves your ability to solve all kinds of logic puzzles. You'll be surprised how many people who didn't study math at university will get the answer to this puzzle wrong:

You are given cards with only a letter on one side and only a digit on the other. Your task is to flip as few cards as possible when checking whether it is true that if there is an odd number on one side, there is a vowel on the other. The card faces you see are: R 8 E 3

Which cards do you need to flip?

If you studied math at all, however, you get the answer right instantly.

Answer 9

The standard test for distinguishing mathematicians from normal people is a two-part test. In Part One, there is a kettle full of water on the floor, and a stove with one burner lit: how do you heat the water in the kettle? Everyone answers, pick the kettle up off the floor, and put it on the lit burner.

In Part Two, there is a kettle of water on a table, and a stove with one burner lit: how do you heat the water in the kettle? If you answered, take the kettle off the table, and put it on the lit burner, well, there's nothing wrong with that answer, but it does prove you're not a mathematician. The mathematician's answer is,

take the kettle off the table, and put it on the floor. This reduces it to a problem already solved.

Answer 10

There is also Hamiltonian Path problem from the Graph Theory. Although I could not find a name for this question, I can explain it in a detailed way:

Hamilton, the prisoner, is in the $5 \times 8$ dungeon. Can the prisoner start at $A$ and end at $B$, visiting each of the $40$ cells exactly once by moving upward, downward, leftward and rightward only?

Moreover, it also can be asked that in which conditions, this can be done? By conditions I mean, for example we have $m \times n$ dungeon and for which $(m,n)$ pairs, prisoner can go from upperleft corner to lowerleft corner by visiting each of $mn$ cells exactly once?

This was famous when I was in primary school and it was shown us as a "mind game". However, a good mathematician can solve this with a simple argument and can reach a general result easily using the same argument:

For the specific case of $5 \times 8$, it is not possible because in total, there are $39$ cells that prisoner should visit (because he visits $A$ in the beginning). Now assume that $u$ is the total upward moves that prisoner does, $d$ is the downward, $r$ is the rightward and $l$ is the leftward moves, similarly. Then we need to have $u+d+r+l = 39$ (since he should visit each cell exactly once). But also notice that when he finishes his travel, net displacement of him is $4$ cells downward and $0$ cell rightward-leftward direction. Therefore, we have $d-u = 4$ and $r = l$. Now it is easy to see that since $d-u$ is an even number, $d+u$ is also even and since $r = l$, $r+l$ is also an even number. But $u+d+r+l = 39$ is an odd number so he can't have a way to travel each cell exactly once, starting from $A$ and ending at $B$.

Answer 11

The Monty Hall problem probably fits the bill.

Answer 12

Anything with statistics and probability throws off the general public.

For example, the question:

Does men have more sex partners than women, on average?

Most mathematicians should figure this out, but non-mathematicians might go with the incorrect gut feeling.

It is generally believed that men are more promiscuous than women, so most people would say yes. However, (assuming no same-sex relations), it is clear that every new sex partner for a guy, also implies a new partner for his female partner, thus keeping the averages exactly the same.

Answer 13

There are more rational numbers than natural ones? And what about real numbers? (Cantor's diagonal argument)

Can you prove there are 2 people in the UK with the same number of hairs? (Pigeonhole principle)

Answer 14

Use six pencils to form four equilateral triangles each with sides a pencil length.

If this is a bar bet, use toothpicks rather than pencils.

Answer 15

I have found that what really confuses people who have not studied mathematics is questions that involve maturity of chances.

A good example of this is the Monte Carlo Fallacy. It basically states that "Red has come up 10 times in a row on this roulette, the next time it is bound to be black". A mathematician of course knows why this doesn't work, but as the casinos in Las Vegas show a large enough portion of people don't know this, or choose to ignore this.

Answer 16

Taken from Puzzling, all credits to NL628: link

Suppose you have 100 lbs of cucumbers and these cucumbers consist of 99% water. You decide to leave the cucumbers in the sun for a while until they consist of 98% water. You bring the cucumbers back in, and you think, "Now the cucumbers should weigh a little less than they were before, right?" But, you try as hard as you can, and you still can't figure out how much they weight. How much do the cucumbers weigh?

Solution discovered by Votbear:

I'm gonna say:

50 lbs

Explanation:

(Assuming the 99% water is by weight)

- The % of X in the cucumbers is calculated as Weight of X / Weight of the cucumbers.

- In the start there's 99% water and 1% solids in 100 lbs of cucumbers.

- Only the Weight of water will change in the process. The Weight of solids won't change after evaporation.

- Going from 99% water to 98% water means the % of solids doubled from 1% to 2%.

- Recall that % of solids = Weight of solids / Weight of cucumbers.

- Since the % of solids was doubled, and Weight of solids didn't change, that could only mean the Weight of cucumbers is halved.

As such, the remaining total weight is 100 / 2 = 50 lbs.

Answer 17

I think a question like

Does the set of "all sets that do not contain itself" contain itself?

would more or less divide people that know set theory foundations from anyone else (based on how someone responds), though I feel this is more of a "trick question" approach.

Answer 18

I have two boxes: box A and box B. I place \$1 bills into box A. For each \$1 bill placed int A, I place a \$10 into B. Suppose I do this infinitely many times. Which box contains more value?

Answer 19

When I was an undergrad, young and brash, I would ask in the first lecture of every math class, “Is this where we finally learn why 1 + 1 = 2?” Before I got to Peano arithmetic, one of my classmates finally said to me, “You do know that 1+1 is defined as 2?”

(At least, that’s how I remember him saying it. Interesting discussion in the comments about whether it’s better to define 2 as 1+1.)

Answer 20

A really simple question is "What's larger: 32% of 25, or 25% of 32?"

A general strategy for finding examples would be to look at "real world" applications of mathematical fields. Combinatorics is an area with a lot of such problems. So, for instance, "How many ways can the letters of 'puppies' be arranged in different sequence of letters?" should qualify.

For real analysis, an example would be "If someone walks up a hill starting at 1:00 PM and ending at 2:00 PM, and walks down from 2:00 PM to 3:00, is there any time at which they were at the same point, separated be exactly one hour?"

For group theory, there are a lot of examples based on the orbits of group actions. For instance, "You have N objects in a circle. Is there a sequence of moves that reverses the order of the objects, if each move consists of picking up one object, sliding the next two back one space each, and then putting the object you picked up back in the space you opened up?" Or "Suppose you have a set of non-zero numbers. You can replace any number with the difference between it and another number. Is there a way to set all the numbers to zero?"

For statistics: "Do men or women have the largest average deviation from the mean height? That is, if for each man you were to give them a positive or negative number representing how much taller or shorter they are than the average man, and do the same for women, which group would have the larger average such number?"

Answer 21

The one that is understandable for both has to do with the well-known "fly between two trains" problem. Especially the variant given in this post.

Two trains 150  miles apart are traveling toward each other along the same track. The first train goes 60 miles per hour; the second train rushes along at 90 miles per hour. A fly is hovering just above the nose of the first train. It buzzes from the first train to the second train, turns around immediately, flies back to the first train, and turns around again. It goes on flying back and forth between the two trains until they collide. If the fly's speed is  120 miles per hour, how many times does it touch the trains?

I have had a long discussion about this with my friends and they still don't believe me. One reason might be that it is indirectly assumed the fly is just a point (it does not have a volume). Even after explanations about that they still say that the number must be finite without a proof.

Answer 22

How can we cut a hemisphere, by a plane parallel to its base, into two parts of equal curved surface areas?

We just need Archimedes' Hat-Box Theorem. The distance between the cutting plane and the base of the hemisphere is equal to half the radius of the hemisphere.

Answer 23

Example 1:

This question never fails to confuse non-mathematicians. It is pretty good at confusing even beginner math students, people who were "good" at math in high school.

You have a summer vacation home by a lake and through the back window you can only see exactly one-half of the lake. One summer, some algae starts growing on the surface of the lake where you can't see it from the window. The surface area of the algae doubles every day such that in thirty days it will cover the entire lake. How many days before you'll see the algae through your window?

The unstated assumption is that I am asking for the worst case scenario so the algae starts at the point furthest away from the visible part of the lake. The solution is:

You will not the see the algae until day 29.

People are shocked and amazed because people love linear relationships and just assume linear relationships by default. In this case, I tell them it is doubling everyday so it is a geometric relationship but everyone still says something like 15 days. Or sometimes they realize it is a "trick" question and the answer can't be as obvious as 15 so it is 14 or 16 or something. Anyone who is decently trained to listen carefully and think critically should get this after about five seconds.

I always use this example whenever I am talking about exponential growth/decay, especially when I am explaining global warming, population growth, depletion of resources, polar ice caps melting, etc. The moral of the story is, sometimes we will simply not even see the problem until it is too late.

Example 2:

Another category which always confuses the general public is percentages. I have managed to confuse small business-owners, cashiers, and salesmen who deal with buying, selling, discounts, taxes, sales, markups, growth rates, profits, and losses every single day. The biggest shock for me was someone who had been in business for more than a decade. He was confused and couldn't give me a correct answer despite repeated attempts, with a calculator.

You walk into a store and purchase an item. The total at the counter, including sales tax, was \$100. The sales tax rate is 10%. What was the tag price of the item on the shelf?

The cultural assumption is that the displayed tag price of the item doesn't include the sales tax so sales tax is added at the counter. I know that different parts of the world have different norms but this is how it is done in the USA.

Inevitably the first, and the intuitive, answer is always \$90. Then I point out that \$90 plus 10% gives you \$99 so that cannot be the correct answer. After the shock and the amazement, they think I am bamboozling them with the language. I point out that it is a straightforward, simple, and short question. It is not a trick question. They always ask me to repeat and they listen super carefully the second time. Then the (cop-out) answer is that "oh, it is just a bit more than \$90". I say yes but can you tell me exactly what it is. Then they grab a calculator and start incrementally increasing \$90 and adding ten percent to it until they get \$100. Some are quick enough to use a bisection-method-variant to quickly hone in on the correct amount in a couple of tries.

Anyone who remembers beginner algebra should quickly setup the equation $x+0.1x=100$ and get the original tag price to be \$ 90$\frac{10}{11} \approx \$90.91 $. This results in the total amount being \$100 to two decimal places.

The problem here is of course, that people always assume 10% of the final amount, which is \$100. We don't want that. The sales tax is calculated on the tag price and then added to get the final price. The final price should be divided by 1.1 instead of multiplied by 0.9. The reciprocal of 1.1 is not 0.9. Adding ten percent of a quantity and then subtracting ten percent of the sum will not give you the original quantity. You will subtract a tiny bit more than you should.

Sure, for a mathematician this is not mathematics. It is just arithmetic. But for the general public this is indeed "math".

Example 3:

Another one, which I always give as extra credit, confuses even advanced undergrads.

Give me two numbers which add up to ten and multiply out to a hundred.

The unspoken rule here is that don't assume numbers to be the integers only, positive or otherwise. This is a super short, easy to understand question.

Everyone starts with thinking something like ten times ten is a hundred but ten plus ten is twenty. Okay, not quite, but the answer should be close enough to ten and ten. Maybe something like eleven and nine. Then they start increasing one, decreasing the other. Some people start systematically writing tables. But of course, there is no solution in the integers.

A mathematician, after trying a few integers as above, should setup the system, \begin{array}{cr} x+y&=&10\\ xy&=&100 \end{array} obtaining the equation $x^2-10x+100=0$ which gives you \begin{array}{cl} x&=&5+i\sqrt{75}\\ y&=&5-i\sqrt{75}. \end{array} These indeed add up to ten and multiply out to one hundred but they just happen to be complex numbers.

The problem here is the "number" assumption. Upon hearing the word "number" everyone assumes that "number" means a positive integer. A mathematician should always think about the assumptions that the question or the answer statement assumes. Okay so maybe this is a "trick" question. A bit disingenuous to ask the general public but math undergrads should always get this. It is my favorite extra credit problem to give.

Answer 24

If you increase a value by $10\%$, then increase again by $10\%$, what is the final increase?

Most of the people would respond $20\%$.

But $a\times1.1\times1.1 = a\times1.21$. Hence the final increase is $21\%$.

Answer 25

Rotate a sphere eastward by ninety degrees. Then rotate it forward by another ninety degrees. The net result is a rotation around some axis. The question is: by how many degrees? (Note: this question can be answered without using paper and pencil. Also, what we concern here is the net change. A rotation by 361 degrees is regarded the same as a rotation by 1 degree, or by 359 degrees if you consider the opposite clock direction.)

I don't know how a non-mathematician would approach it. Based on my experience, a non-mathematician (or strictly speaking, a person without mathematical training at undergraduate level) may even have trouble understanding why the composition of two rotations is still a rotation.

Answer 26

The pigeonhole principle is probably a rich source for generating questions which meet your criterion.

Suppose that you're in a 2 meter by 2 meter room and you throw five pennies in the air. After they land is it possible that each penny is further than 1.5 meters from its nearest neighbor?

Answer 27

When I was in high school, I was given a math test that included the following problem:

A school has a line of 100 lockers labeled 001 through 100, all of which are closed. The first student goes by and opens every locker. The second student goes by and closes every other locker. The third student goes by and for every third locker, they open it if it was closed or close it if it was open. The fourth student goes by and for every fourth locker, they open it if it was closed or open it if it was open. This process continues, with the Nth student going to every Nth locker and opening it if it was closed and closing it if it was open.

After 100 students have gone by in this way, which lockers are open and which are closed?

.

A mathematician will quickly see that lockers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are open. They do this by observing that the Nth student will only toggle a door if there is a number A such that N*A is the number on the locker. This means there must also be a student A which also toggles the same locker, canceling out the first student's work. The exception to this are the numbers which are squares, where there will be a N*A where N==A, and thus there will be no other student to close the door.

Since I'm an engineer, not a mathematician, I suppose this means that the test doesn't prove you are a mathematician, but I also geek out over the Banach–Tarski paradox, so maybe I'm just weird.

Answer 28

1) Brower fixed point in its "coffee cup" formulation: prove that when you put sugar in a coffe and then mix with a spoon, at every moment there is a gut of coffee that did not moved.

1.1) Also the hairy ball theorem could work

2) Prove that there exists a real number $x$ such that $x^3=2018x^2+8021x+3$

3) Prove that if you put the map of your city on the table then there is a (unique) point on the map which is exactly in its real postion (contractions theorem)

4) in how many ways one can arrange 5 objects on a line?

4.1) in how many ways one can choose 3 of 5 objects?

5) An annulus can be deformed to a cylinder?

6) Two polygons with the same perimeter do they have the same area?

Answer 29

The red stripe on a barber pole makes one complete revolution around the pole. If the pole is 50 inches tall and has the amazingly precise radius of $\frac{25}{π\sqrt{3}}$ inches, what angle does the stripe make with base of the pole, and how long is the stripe? (Assume the stripe is a thin line.)

In my experience, someone with mathematical training is much more likely to think of "unrolling" the barber pole to make a rectangle, with the stripe as a diagonal. A mathematician also has a good chance of noticing that this rectangle has height 50 and width $\frac{50}{\sqrt{3}}$, and realizing that the diagonal makes a 30-60-90 triangle with the sides.