[Math] What kid-friendly math riddles are too often spoiled for mathematicians

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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

Best Answer

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?
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