math problem solving and hard math problems

3 Math Problems to Solve the Logical Way

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Sometimes math delves into the realm of logic, where thinking overtakes number crunching. Logic is what operates computers, coding algorithms, search engines, and other man-made devices. When it comes to solving math problems that are rooted in logic, your thinking approach has to be sharp. Here are three logic problems and some guidance on how to properly solve them:

Ping-Pong Ball in a Hole

The scenario: you’re playing ping pong with a friend, and suddenly the ball falls down a pipe in the ground. The pipe looks to be like it’s buried 2 feet into the ground, and you can see the ball. All you have are the ping-pong paddles, a roll of tape, and a half-empty water bottle to pull the ball up. What do you do?


In order to solve this problem, it takes a little bit of seeing through misdirection. You don’t even need any tools given to you in the process. Just drain your water bottle into the pipe in the ground, and the ping-pong ball should easily float to the top. If you’ve already drunk all of your water, well we’re sure you can find a way to get the ball to the top anyhow.

The Right Fingers of the World

Let’s say the world is populated with approximately 7 billion people (which it is, last we checked). What do you think the resulting number would be if you multiplied all the right fingers on everyone’s right hand together? Yes, thumbs too. How high of a number could that be?


This kind of problem solving is a little tricky. Of the 7 billion people in the world, there’s at least one that has zero fingers on their right hand. We feel sorry for those individuals, but the result of multiplying anything by zero, regardless of the previous operations or numbers, is still zero. Gotta think different, sometimes!

100 Dragon Eyes Logic

This has been circulating the interwebs for awhile. We recently found this months ago and decided it w0uld be a good idea to re-circulate it here. Considered one of the most difficult math problems to solve, this is the Green-Eyed Dragon Logic Puzzle:

You visit a remote desert island inhabited by one hundred very friendly dragons, all of whom have green eyes. They haven’t seen a human for many centuries and they are very excited about your visit. They show you around their island and tell you all about their dragon way of life (dragons can talk, of course).

They seem to be quite normal, as far as dragons go, but then you find out something rather odd. They have a rule on the island which states that if a dragon ever finds out that he/she has green eyes, then at precisely midnight on the day of this discovery, he/she must relinquish all dragon powers and transform into a long-tailed sparrow. However, there are no mirrors on the island, and they never talk about eye color, so the dragons have been living in blissful ignorance throughout the ages.

Upon your departure, all the dragons get together to see you off and in a tearful farewell you thank them for being such hospitable dragons. Then you decide to tell them something that they all already know (for each can see the colors of the eyes of the other dragons). You tell them all that at least one of them has green eyes. Then you leave, not thinking of the consequences (if any).

Assuming that the dragons are (of course) infallibly logical, what happens? If something interesting does happen, what exactly is the new information that you gave the dragons?

In order to solve this problem, you have to bring it to the basics. Let’s examine this further. If the island was inhabited by one dragon and “you” inform the dragon before your departure that “at least one dragon has green eyes,” then this lone dragon would certainly turn into a sparrow at night.

But, what if there were two?

If two dragons inhabited the island, and once again you inform the dragons that “at least one dragon has green eyes,” then both dragons would look at each other and recognize the other dragon would have green eyes and they would assume the other dragon would turn into a sparrow at midnight. Since neither do as they don’t know the color of their eyes, they turn into sparrows the second night at midnight.

This kind of math problem solving is difficult to think about and doesn’t make sense at first, but if you stick to strict logic, then every dragon will turn into a sparrow after 100 nights on the stroke of midnight. It’s hard to warp your mind around, we know, so if you need some further reading material on solving this problem, we recommend reading XKCD’s take on the problem. It’s worded differently, but the premise is the same.

Thanks a ton for reading! If you’re interested in reading about more hard math problems, take a look at other blog posts we have, or contact us if you’d like to learn more about the Areteem Institute.

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