What kid-friendly math riddles are too often spoiled for mathematicians?

Question

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

Answer 1

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?

Answer 2

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."

Answer 3

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

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Answer 4

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?

Answer 5

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.

Answer 6

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.)

Answer 7

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?

Answer 8

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?

Answer 9

• 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?)

Answer 10

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.

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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.

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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.

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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