46
$\begingroup$

Some math riddles tend to be spoiled for mathematicians before they get a chance to solve them. Three examples:

  1. What is $1+2+\cdots+100$?

  2. Is it possible to tile a mutilated chess board with dominoes?

  3. Given a line $\ell$ in the plane and two points $p$ and $q$ on the same side of $\ell$, what is the shortest path from $p$ to $\ell$ to $q$?

I would like to give my children the opportunity to solve these riddles before the spoilers inevitably arrive.

Question: What are other examples of kid-friendly math riddles that are frequently spoiled for mathematicians?

Notes:

  • There is no shortage of kid-friendly math riddles. I am specifically asking for riddles that are frequently spoiled for mathematicians because they capture a bigger idea that is useful in math, especially research-level math. As such, the types of riddles I am asking for are most readily supplied by research mathematicians.

  • In case it is not clear whether MO is an appropriate forum for this question, see the following noteworthy precedent: Mathematical games interesting to both you and a 5+-year-old child

$\endgroup$
16
  • 14
    $\begingroup$ I'm not sure if being aware of the bigger mathematical perspective counts as having it 'spoiled' …. Do you include the famous von Neumann "what is easier than summing the series?"? $\endgroup$
    – LSpice
    Feb 23 at 15:13
  • 16
    $\begingroup$ You are asking in the wrong forum, I think. $\endgroup$ Feb 23 at 15:19
  • 8
    $\begingroup$ This is an interesting question, and I'm really curious to see the answers proposed, but it might be better received in math.stackexchange. $\endgroup$
    – JoshuaZ
    Feb 23 at 15:38
  • 7
    $\begingroup$ Try Martin Gardner's book My Best Mathematical and Logic Puzzles. Not all of these are suitable for kids but some are. Sometimes he also prefaces the statement of a puzzle with an easier "too often spoiled" puzzle (is there more water in the wine or more wine in the water?) so it's also a good source for those. $\endgroup$ Feb 24 at 1:26
  • 14
    $\begingroup$ Suppose that $\lambda$ is a singular limit of supercompact cardinals, how many cardinals lie between $\lambda$ and $2^\lambda$? $\endgroup$
    – Asaf Karagila
    Feb 24 at 13:24

10 Answers 10

32
$\begingroup$

Here's a few, two I got to solve myself as a kid and one (a trickier one, in my opinion) that was spoiled for me.

  1. There are $1000$ lights all in a line and turned on. At time $n$, person $n$ comes by and toggles the switch on every $n$th light, starting with the $n$th. How many lights are on after person $1000$ has finished?
  2. There are four ants standing at the corners of a square of side length $1$. At time $t=0$, they begin walking with speed $1$, each toward the ant to their right. How long does it take them to all meet in the center?
  3. There are $20$ soldiers standing distance $1$ apart on a bridge of length $19$. At time $t=0$, the soldiers immediately begin walking left or right with speed $1$. When two soldiers collide, they immediately turn around and begin walking in the opposite direction. What is longest possible time it takes all the soldiers to leave the bridge?
$\endgroup$
18
  • 10
    $\begingroup$ Is the kid friendly way to solve 2 without calculations to talk about infinitesimals (after the symmetry discussion), or is there some "trick" to seeing it that I'm missing? $\endgroup$
    – Ville Salo
    Feb 24 at 5:23
  • 12
    $\begingroup$ @VilleSalo By symmetry the ants are going to always stay arranged in a square. For any given ant in a square arrangement, the ant it's moving towards will always be moving directly sideways, and sideways movement doesn't affect separation distance. The only thing affecting distance between the two ants is the first ant's walking speed, so the distance between them shrinks at constant speed 1. The distance starts at 1, so at speed 1 it drops to 0 at t = 1. $\endgroup$
    – Douglas
    Feb 24 at 8:39
  • 9
    $\begingroup$ Yes, and I was wondering if there's a way to make the "moving sideways" argument more "rigorous"/convincing: maybe a kid will find it obvious, but if they don't, what do you do? The issue is, if you think in terms of discrete moves, it may not be so clear, and as mentioned in my comment above, I think I once managed to convince myself (I was not really given this as a puzzle, I was just told the wrong solution) that actually the ants do not get closer even though at any given moment it looks like they do. $\endgroup$
    – Ville Salo
    Feb 24 at 9:23
  • 7
    $\begingroup$ My understanding is that it took humankind thousands of years to sort these infinitesimal issues out, but sometimes there is a convincing geometric trick, and I was wondering if for example there's something like that here. $\endgroup$
    – Ville Salo
    Feb 24 at 9:25
  • 4
    $\begingroup$ Are the lights on or off before the 1st person comes? $\endgroup$ Feb 24 at 10:26
26
$\begingroup$

To make this suitable for MO rather than math.SE, perhaps we can define a "too often spoiled" puzzle to be one that can be recognized instantly by a mathematician even with what looks like far too little description. So for example, the following "words to the wise" should be sufficient in each case (some of these have already been mentioned by others):

  1. It's dark and you have ten white socks and ten black socks in your drawer.

  2. Is there more water in the wine or more wine in the water?

  3. A fox, a rabbit, and a cabbage.

  4. How do you measure out exactly 5 gallons?

  5. Four people are crossing a bridge.

  6. You come upon a fork in the road.

  7. A checkerboard is missing two squares.

  8. Von Neumann said, "I summed the series."

  9. There's a rope around the equator of the Earth.

  10. There are three doors.

  11. What is the probability that my other child is a girl?

  12. There are 12 coins, one of which is lighter or heavier than the others.

  13. You arrive on an island where some people have blue eyes.

  14. "I don't know the numbers." "I don't know the numbers." "Now I know the numbers." "So do I."

$\endgroup$
20
  • 3
    $\begingroup$ @EmilJeřábek Okay, then, a mathematician with a strong interest in recreational mathematics. $\endgroup$ Feb 24 at 15:05
  • 19
    $\begingroup$ "Each prisoner is wearing a hat" should be in that list, I believe. $\endgroup$ Feb 24 at 15:41
  • 13
    $\begingroup$ A bat and a ball cost $1.10 in total. $\endgroup$
    – Ira Gessel
    Feb 24 at 17:01
  • 4
    $\begingroup$ A hiker climbs up a mountain, spends the night at the top, and climbs down the next day. $\endgroup$ Feb 24 at 22:15
  • 2
    $\begingroup$ Worst list of opening lines, ever. Excepting submissions to the Bulwer-Lytton fiction contest. $\endgroup$ Feb 25 at 17:57
11
$\begingroup$

The shortest path of a fly walking on the interior surface of a cubic room:


      Image credit


$\endgroup$
9
  • 15
    $\begingroup$ Upon reflection [sic] this isn't very different from the OP's third examples (shortest path from point $p$ to line $\ell$ to point $q$ on the same side). $\endgroup$ Feb 24 at 0:57
  • 6
    $\begingroup$ Quanta recently published a very nice article about this kind of thing recently: quantamagazine.org/the-crooked-geometry-of-round-trips-20210113 The question is: does every shortest round trip from a vertex to itself passes through another vertex? It's true for any platonic solid (try to prove it for the tetrahedron, it's pretty fun)... except the dodecahedron! (Not affiliated with quanta, I just found it rather interesting!) $\endgroup$ Feb 24 at 8:22
  • 2
    $\begingroup$ @NajibIdrissi - I think there may be special uses of shortest and round trip involved there. For example there are trips on a tetrahedron which visit every face and return to the original vertex shorter than any trips which visit another vertex and return. And on a dodecahedron, even restricted to a straight line on a plane from unfolding/tumbling, I would guess that the shortest round trip actually visits five other vertices. $\endgroup$
    – Henry
    Feb 25 at 0:13
  • 1
    $\begingroup$ @NajibIdrissi I did - both the Quanta article and the two papers on arXiv. Which is why I know shortest is not used in its conventional sense. $\endgroup$
    – Henry
    Feb 25 at 9:53
  • 1
    $\begingroup$ @NajibIdrissi I am aware of that (and what happens crossing a vertex is even more complicated as there are choices involved) but one of my points is that a child could understand shortest as minimum total distance, and that minimum trip for a dodecahedron passes through other vertices. $\endgroup$
    – Henry
    Feb 25 at 10:06
7
$\begingroup$

The book "1000 Play Thinks" by Ivan Moscovich contains up to 1000 of these, depending on your background. It is an absolute delight - large pages, full-coloured and playfully illustrated by Tim Robinson. Puzzles are grouped by mathematical categories (Geometry, Graphs and Networks, Numbers, Probability, Topology...), show essential examples, structures and ideas from those fields, and each has a difficulty rating and solution. Between puzzles are short introductions to subjects and historical notes of the mathematicians involved in their development. It also includes 89 references to other mathematical puzzle books.

Flipping through various sections, here are a few examples:

  • 38: Will a $70$ cm sword fit into a $30\times 40 \times 50$ cm chest?
  • 179: Euler's Problem: "to trace a pattern without picking up your pencil or backtracking over sections." Along with $11$ images and the question "which ones do you find impossible to solve?"
  • 186: Utilities I: Can you connect three house to three utilities without allowing any of the lines to intersect? Followed up by three Play Thinks on multipartite graphs (including the terminology, and phrased as connecting animals of various colours).
  • 528: A description of perfect numbers, the example of 6, and the question: what is the second perfect number? Also notes that 38 perfect numbers are known, so the book is dated between 1999 and 2001.
  • 687: You need to roll a double 6 in at least one of twenty-four throws. Are the odds in your favor?
  • 703: Mars Colony (Gerhard Ringel's "Empire-Colony puzzle" of colouring two maps with 11 numbered regions so that both regions with the same number have the same colour.)
  • 715: Topology of the Alphabet. Can you find the letters that are topologically equivalent to E in the given font?
  • 859: A steel washer is heated until the metal expands by 1%. Will the hole get larger or smaller or remain unchanged?
  • 995: Seven birds live in a nest, and send out three each day in search of food. After 7 days, every pair of birds has been one one foraging mission together. Can you work out how?
$\endgroup$
1
  • 1
    $\begingroup$ Oh, right, the Konigsberg bridge problem (special case of 179) is one that I had spoiled for me. $\endgroup$ Feb 26 at 0:50
6
$\begingroup$

A new family moved into your neighborhood. You heard that they have two children, but don't know if boys or girls. You look out the window, and see a girl playing outside that you never saw before. So one of the new children is a girl. What is the probability that the other one is a girl?

Edited, to address the comments:

A new family moved into your neighborhood. You heard that they have two children, but don't know if boys or girls. You meet the parents and ask if both children are boys. They answer no, so you know that at least one of them is a girl. What is the probability that the other one is a girl too?

I used it once in a talk to an undergraduate math club. After arguing with the audience we ran a simulation, with coins instead of children. Demonstrated that math was right and common sense was wrong.

$\endgroup$
20
  • 5
    $\begingroup$ The Monty Hall puzzle is a famous variant of the same idea. $\endgroup$ Feb 24 at 7:45
  • 14
    $\begingroup$ I think the way you phrased the problem with the children actually does have 1/2 as the answer. You didn't learn "one of the children is a girl". You learned "The child outside is a girl". Normally you'd have 4 options (2 times boy/girl) and eliminate the boy+boy one to get 1/3. But here you have 8 options (2 times boy/girl and what child did I see). And by seeing a girl you eliminate 4 out of 8. $\endgroup$
    – Jorik
    Feb 24 at 10:54
  • 3
    $\begingroup$ I agree with Jorik. A different way of phrasing without extraneous details could be: "someone tosses two coins at the other side of the room, so that you don't see the result. The person names, at random, one of his two results. What is the probability that his other result is different then the named one?" Its not so clear which formulation better describes the situation, and they have different answers. I dont think its fair to use this to illustrate how common sense is wrong, but merely how subtle probability is to formulation of the problem. $\endgroup$
    – Amueller
    Feb 24 at 11:13
  • 2
    $\begingroup$ @GregoryArone The issue with someone spontaneously saying "At least one of the coins came up heads" is that you can only calculate the probability if you assume they are equally likely to say that regardless of if one or two came up heads. It's easy to come up with potential reasons why that wouldn't be so. So you really want a scenario where someone asks the question. I don't think this gives the game away - someone who understands how to get the solution in this scenario understands the key idea. For the riddle, I think the best approach is to justify it. $\endgroup$
    – Will Sawin
    Feb 24 at 13:36
  • 2
    $\begingroup$ e.g. Perhaps you have boy's clothes that your son grew out of and want to know if one of the children is a boy in case they want hand-me-downs (this isn't ideal because there is overlap in clothes that children of different genders may want to wear, but it's the best I could come up with.) $\endgroup$
    – Will Sawin
    Feb 24 at 13:43
6
$\begingroup$

Given an equilateral triangle with side length 1 and five points within that triangle's interior, some pair of those points is at a distance less than $\frac{1}{2}$. (Or other similar problems using

the pigeonhole principle

.)

Not sure if this following one is one that's commonly seen, but it's possibly a bit more kid friendly: Out of any list of ten integers, there is some nonempty subset whose sum is divisible by 10. (And then, you see you can in fact make it some consecutive nonempty subset of the integers, assuming some ordering to the initial list.)

$\endgroup$
2
$\begingroup$

Very similar to the shortest path ones already mentioned: A person is on the shore at a perpendicular distance $d_1$ from the shore and wants to rescue a person drowning at a perpendicular distance $d_2$ from the shore and a distance $\ell$ from the first person's position on the shore (usually that's sketched); they can run at speed $v_1$ and swim at speed $v_2$. What path should they take to the drowning person in order to rescue them as quickly as possible?

$\endgroup$
2
$\begingroup$

The Monty Hall Problem is a famous one, which goes against most people's intuitions.

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

$\endgroup$
3
  • 2
    $\begingroup$ It's crucial to understanding the problem to know that the host always reveals what's behind another door and offers the chance to switch (which I didn't know the first time I read the problem, never having seen the show). If the host has a choice instead to just give you what's behind your chosen door immediately, the answer could be different depending on host's strategy. For example, if the host's strategy is: if the person chose correctly show and give choice to switch, if the person chose incorrectly open chosen door - then you never want to switch. $\endgroup$ Feb 26 at 0:48
  • 1
    $\begingroup$ One also needs to know whether the host is allowed to open the door with the car behind it and show you that you lost! $\endgroup$
    – Dan Ramras
    Feb 26 at 2:14
  • 1
    $\begingroup$ I am hopelessly math avoidant. I never really understood it in high school, and that was many decades ago... This problem was discussed in the movie 21, and I didn't understand it there, nor can I figure it out now. Sometimes I hate how illiterate I am with any form of higher math, beyond the basics (add, sub, mult, div). $\endgroup$
    – CGCampbell
    Feb 26 at 18:08
-3
$\begingroup$
  • Can you find two different (real) numbers with nothing in between them?
  • If you're at the beach, and the coastline's wobbly, how can you find the shortest distance to the water?
  • If you connect up the midpoints of the sides of a triangle, how much smaller is the triangle you get? (What about a square, or a pentagon?)
$\endgroup$
2
  • 5
    $\begingroup$ How is that a kid-friendly riddle? $\endgroup$
    – gmvh
    Feb 24 at 13:19
  • $\begingroup$ @gmvh It might not be, but it's usually spoiled in childhood. It's certainly fun to mess around with – though you probably wouldn't say “real” to the child. If they come back with 1 and 2, give them 1½, and that should be enough. $\endgroup$
    – wizzwizz4
    Feb 24 at 13:21
-6
$\begingroup$

I) The game of the little pîglet is a French game for 3 players. The first player takes the ball and throws it. The second player takes the throw and balls it. What does the third player do?

Used to illustrate first order logic, as it is NOT a guessing game, the correct answer is the only answer which can be logically deduced from the riddle.


II) Ann goes out with some hard boiled eggs in her basket (If you forget "hard-boiled", the child will point out to you that it is impossible). She meets her sister and gives her half the eggs in her basket plus half an egg. She meets her cousin and gives her half the eggs in her basket plus half an egg. She meets her sister-in-law and gives her half the eggs in her basket plus half an egg. She then goes home with no more eggs. How many eggs did she have to start with?

Spoiled if solved via algebra.


III) There are 100 red buttons and 99 black buttons in a large jar. A kid takes them out two by two, and he has some to spare (enough for the whole game). He cannot see which buttons he takes out, the jar is opaque. When he draws two buttons of the same colour, he puts a red button back in the jar. When he draws two buttons of different colours, he puts a black button back in the jar. When only one buttons is left in the jar, which colour is it?

Spoiled by probabilities - and that's overkill. You have to ask why.


IV) For kids only starting geometry, ask them to prove the Pythagorean Theorem on the special case where you start with half a square, to prove it the ancient way, using only a straight stick, a pen and a compass. For older kids, ask them the same starting with half a rectangle.

Spoiled by the modern mathematical proof

$\endgroup$
15
  • 13
    $\begingroup$ (2)–(4) are very nice examples, but (1) seems incomprehensible (and searching, I can’t find an answer anywhere else). Can you give a ROT13’d hint, or a link to an answer, or some other non-spoiling form of help? $\endgroup$ Feb 24 at 10:23
  • 4
    $\begingroup$ In what sense is the third one spoiled by probabilities? First, red herrings and overkill are different things, and second, having seen a different problem where the solution is a similar trick before is not really the same as having the problem spoiled. Also, rot13 is a simple form of "encryption" that is sufficient to prevent someone from accidentally seeing the answer if they don't want to but is easy to decrypt if you do want to see. en.wikipedia.org/wiki/ROT13 rot13.com $\endgroup$
    – Will Sawin
    Feb 24 at 13:30
  • 7
    $\begingroup$ For (I), a few webpages (all written in French) suggest that rot13(guvf evqqyr vf haqrefcrpvsvrq, naq gung'f gur cbvag). Considering this riddle isn't well known to the MO community, I'm not sure this qualifies as "too often spoiled for mathematicians." $\endgroup$ Feb 24 at 14:47
  • 8
    $\begingroup$ @DustinG.Mixon If that’s right, then I fail to see what it has to do with first-order logic. Anyway, the usual meaning of the word “illustrate” is to “clarify with a helpful example”. Here, it seems to be just the opposite. $\endgroup$ Feb 24 at 14:55
  • 3
    $\begingroup$ The French versions of the first riddle I find online are based around the fact that "le jeu se joue à trois" can mean both "is a game for three players" and "the game starts on the count of three". I don't see the link with first order logic, so this must be a different riddle. $\endgroup$
    – Tassle
    Feb 25 at 12:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.