Tag Archives: math problem

Solutions to Our June Brain Teasers!

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Two weeks ago, we shared a pair of brain teasers sent in by a PuzzleNationer who discovered these particular deduction and math thinkers in a book of riddles and puzzles.

Today, we’re going to share not only the solutions, but how we got there! Please enjoy this brief solve and tutorial, courtesy of brain teasers from your fellow PuzzleNationers!

[Image courtesy of SharpBrains.com.]

Brain Teaser #1: There is a three-digit number. All three digits are different. The second digit is four times as big as the third digit, while the first digit is three less than the second digit. What is the number?

Solution: 582

This is a fairly simple one, but if you’re unfamiliar with brain teasers, or uncomfortable in general with number puzzles, it can be off-putting. No worries, though! We’ve got you covered.

We know the second digit is four times as big as the third. That leaves only two options for those digits: 4 and 1 or 8 and 2.

If the first digit is three less than the second digit, it can’t be 4 and 1, because that would be 4 minus 1, or 1 for the first digit, and the first and third digits can’t be the same.

That means it’s 8 and 2 for the second and third digits. So if the first digit is three less than the second, the first digit is 5, and the three-digit number is 582.

calendar pages

Brain Teaser #2: When asked about his birthday, a man said, “The day before yesterday, I was only 25, and next year I will turn 28.” This is true only one day in a year – what day was he born?

Solution: He was born on December 31st and spoke about it on January 1st.

The wording in this one is especially important, because at first glance, this sounds impossible.

“Next year, I will turn 28.”

But if you look at the key word in what the man says — “turn” — the puzzle starts to unravel.

If next year, he will turn 28, this means that, at some point this year, he will turn 27. Which means he is currently 26.

Let’s look at what we know:

  • Day before yesterday: 25
  • Currently: 26
  • This year (at some point): 27
  • Next year: 28

Since he’ll be both 26 and 27 this year, the day before yesterday had to be last year.

Which means that yesterday was his birthday.

But at some point this year, he turns 27. That means both yesterday and the day before yesterday had to be last year.

Which leaves us with this timetable:

  • December 30 (day before yesterday, last year): 25
  • December 31 (yesterday, last year, his birthday): 26
  • January 1 (today, this year): 26
  • December 31 (later this year): 27
  • December 31 (next year): 28

He was born on December 31st and spoke about it on January 1st.

Did you unravel one or both of these brain teasers? Let us know in the comments section below! We’d love to hear from you.

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A Conway Puzzle Solution (And Some Hints for the Other Puzzle)

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Two weeks ago, in honor of mathematician and puzzly spirit John Horton Conway, we shared two of his favorite brain teasers and challenged our fellow PuzzleNationers.

So today, we happily share the solution for puzzle #1, The Miracle Builders.

I had a window in the north wall of my house. It was a perfect square, 1 meter wide and 1 meter high. But this window never let in enough light. So I hired this firm, the Miracle Builders, who performed the impossible. They remodeled the window so it let in more light. When when they’d finished the window was a perfect square, 1 meter high and 1 meter wide.

How did they do it?

Both windows are perfect squares, 1 meter wide and 1 meter high. So how can there be a difference in the amount of light?

The trick of this puzzle is in the description. Although the original window was a perfect square, the dimensions of the square aren’t 1 meter by 1 meter. No, it was a square placed like a diamond, with one corner directly above its opposite. So the 1 meter dimensions were the diagonals, not the sides.

All the Miracle Builders had to do was build a square window in the usual arrangement (two sides horizontal, two sides vertical) with dimensions of 1 meter by 1 meter. That creates a larger window (with a diagonal of √2m) and allows more light.

Very tricky indeed.

We had several solvers who successfully cracked the Miracle Builders puzzle, but there was less success with puzzle #2, The Ten Divisibilities.

So, in addition to the original puzzle, we’re going to post some solving hints for those intrepid solvers who want another crack at the puzzle.

The Ten Divisibilities

I have a ten digit number, abcdefghij. Each of the digits is different, and:

  • a is divisible by 1
  • ab is divisible by 2
  • abc is divisible by 3
  • abcd is divisible by 4
  • abcde is divisible by 5
  • abcdef is divisible by 6
  • abcdefg is divisible by 7
  • abcdefgh is divisible by 8
  • abcdefghi is divisible by 9
  • abcdefghij is divisible by 10

What’s my number?

[To clarify: a, b, c, d, e, f, g, h, i, and j are all single digits. Each digit from 0 to 9 is represented by exactly one letter. The number abcdefghij is a ten-digit number whose first digit is a, second digit is b, and so on. It does not mean that you multiply a x b x c x…]

Here’s a few hints that should help whittle down the possibilities for any frustrated solvers:

-If you add all the digits in a number, and the total is divisible by 3, then that number is also divisible by 3.
-If the last two digits of a number are divisible by 4, then that number is divisible by 4.
-If the last three digits of a number are divisible by 8, then that number is divisible by 8.

Good luck, and happy puzzling!

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Two Brain Teasers, Courtesy of Conway

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John_H_Conway_2005_(cropped)

Last week, we penned a post celebrating the life and puzzly legacy of mathematician John Horton Conway, and several of our fellow PuzzleNationers reached out with their own thoughts or questions about Conway.

One recurring subject was about his love of puzzles and what kind of puzzles he enjoyed solving. So, naturally, I went hunting for some of Conway’s favorite puzzles.

As it turns out, Alex Bellos of The Guardian had me covered. Alex has a recurring puzzle feature on The Guardian‘s website where brain teasers and other mental trickery awaits intrepid solvers.

Years ago, Alex had asked Conway for suggestions for his column, and Conway offered up two tricky puzzles.

And now, I happily share them with you.

#1: The Miracle Builders

I had a window in the north wall of my house. It was a perfect square, 1 meter wide and 1 meter high. But this window never let in enough light. So I hired this firm, the Miracle Builders, who performed the impossible. They remodeled the window so it let in more light. When when they’d finished the window was a perfect square, 1 meter high and 1 meter wide.

How did they do it?

#2: The Ten Divisibilities

I have a ten digit number, abcdefghij. Each of the digits is different.

The following is also true:

  • a is divisible by 1
  • ab is divisible by 2
  • abc is divisible by 3
  • abcd is divisible by 4
  • abcde is divisible by 5
  • abcdef is divisible by 6
  • abcdefg is divisible by 7
  • abcdefgh is divisible by 8
  • abcdefghi is divisible by 9
  • abcdefghij is divisible by 10

What’s my number?

[To clarify: a, b, c, d, e, f, g, h, i, and j are all single digits. Each digit from 0 to 9 is represented by exactly one letter. The number abcdefghij is a ten-digit number whose first digit is a, second digit is b, and so on. It does not mean that you multiply a x b x c x…]

Did you solve one or both of these fiendish mind ticklers? Let us know in the comments section below! We’d love to hear from you.

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Celebrating the Puzzly Legacy of John Horton Conway

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The worlds of puzzles and mathematics overlap more than you might think. I’m not just talking about word problems or mathy brain teasers like the Birthday Puzzle or the jugs of water trap from Die Hard with a Vengeance.

For twenty-five years, Martin Gardner penned a column in Scientific American called Mathematical Games, adding a marvelous sense of puzzly spirit and whimsy to the field of mathematics, exploring everything from the works of M.C. Escher to visual puzzles like the mobius strip and tangrams. He was also a champion of recreational math, the concept that there are inherently fun and entertaining ways to do math, not just homework, analysis, and number crunching.

And on more than one occasion, Gardner turned to the genius and innovative thinking of John Horton Conway for inspiration.

John_H_Conway_2005_(cropped)

[Image courtesy of Wikipedia.]

Conway was best known as a mathematician, but that one word fails to encapsulate either his creativity or the depth of his devotion to the field. Conway was a pioneer, contributing to some mathematical fields (geometry and number theory among them), vastly expanding what could be accomplished in other fields (particularly game theory), and even creating new fields (like cellular automata).

Professor of Mathematics, Emeritus, Simon Kochen said, “He was like a butterfly going from one thing to another, always with magical qualities to the results.” The Guardian described him in equally glowing terms as “a cross between Archimedes, Mick Jagger and Salvador Dalí.”

lifep

[Image courtesy of Cornell.edu.]

His most famous creation is The Game of Life, a model that not only visually details how algorithms work, but explores how cells and biological forms evolve and interact.

Essentially, imagine a sheet of graph paper. In The Game of Life, you choose a starting scenario, then watch the game proceed according to certain rules:

  • Any live cell with fewer than two live neighbors dies, as if by underpopulation.
  • Any live cell with two or three live neighbors lives on to the next generation.
  • Any live cell with more than three live neighbors dies, as if by overpopulation.
  • Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction.

The process plays out from your starting point completely without your intervention, spiraling and expanding outward.

It’s the ultimate if-then sequence that can proceed unhindered for generations. It is a literal launchpad for various potential futures based on a single choice. It’s mind-bending and simple all at once. (And you can try it yourself here!)

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[Image courtesy of Sign-Up.To.]

But that’s far from Conway’s only contribution to the world of puzzles.

Not only did he analyze and explore puzzles like the Soma cube and Peg Solitaire, but he created or had a hand in creating numerous other puzzles that expanded upon mathematical concepts.

I could delve into creations like Hackenbush, the Angel Problem, Phutball/Philosopher’s Football, Conway’s Soldiers, and more — and perhaps I will in the future — but I’d like to focus on one of his most charming contributions: Sprouts.

Sprouts is a pencil-and-paper strategy game where players try to keep the game going by drawing a line between two dots on the paper and adding a new dot somewhere along that line.

The rules are simple, but the gameplay can quickly become tricky:

  • The line may be straight or curved, but must not touch or cross itself or any other line.
  • The new spot cannot be placed on top of one of the endpoints of the new line. Thus the new spot splits the line into two shorter lines.
  • No spot may have more than three lines attached to it.

Check out this sample game:

sprouts

[Image courtesy of Fun Mines.]

It’s a perfect example of the playfulness Conway brought to the mathematical field and teaching. The game is strategic, easy to learn, difficult to master, and encourages repeated engagement.

In a piece about Conway, Princeton professor Manjul Bhargava said, “I learned very quickly that playing games and working on mathematics were closely intertwined activities for him, if not actually the same activity.”

He would carry all sorts of bits and bobs that would assist him in explaining different concepts. Dice, ropes, decks of cards, a Slinky… any number of random objects were mentioned as potential teaching tools.

Professor Joseph Kohn shared a story about Conway’s enthusiasm for teaching and impressive span of knowledge. Apparently, Conway was on his way to a large public lecture. En route, he asked his companions what topic he should cover. Imagine promising to do a lecture with no preparation at all, and deciding on the way what it would be about.

Naturally, after choosing a topic in the car, the lecture went off without a hitch. He improvised the entire thing.

Of course, you would expect nothing less from a man who could recite pi from memory to more than 1100 digits? Or who, at a moment’s notice, could calculate the day of the week for any given date (employing a technique he called his Doomsday algorithm).

Conway unfortunately passed away earlier this month, due to complications from COVID-19, at the age of 82.

His contributions to the worlds of mathematics and puzzles, not to mention his tireless support of recreational math, cannot be overstated. His work and his play will not soon be forgotten.

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[Image courtesy of Macleans.]

If you’d like to learn more about Conway, be sure to check out Genius at Play: The Curious Mind of John Horton Conway by Siobhan Roberts.

[My many thanks to friend of the blog Andrew Haynes for suggesting today’s subject and contributing notes and sources.]

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Riddle Me This: Answer Edition

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[Image courtesy of Nyafuu Archive.]

Last week, we shared a sampling of riddles from Raging Swan Press’s series of riddle-filled handbooks, “So What’s the Riddle Like, Anyway?”

And today, we’ve got the answers ready for you. So let’s see how you did!

1. What always runs but never walks,
Often murmurs, never talks,
has a bed but never sleeps,
Has a mouth but never eats?

Answer: a river.

2. What has a head and a tail, but no body?

Answer: a coin.

3. I can be cracked, I can be made.
I can be told, I can be played.

Answer: a joke.

4. What should the tenth number in this series be? 3, 3, 5, 4, 4, 3, 5, 5, 4

Answer: 3. (Each number is the number of letters in the digits one through nine, so ten would be “3.”)

5. A carpenter was in a terrible hurry. He had to work as quickly as possible to cut a very heavy ten‐foot plank into ten equal sections. If it takes one minute per cut, how long will it take him to get the ten equal pieces?

Answer: 9 minutes. (The first 8 minutes yield 8 pieces, but the ninth minute will yield pieces 9 and 10.)

6. Can you find a four‐digit number in which:
The first digit is one‐third the second digit,
The third is the sum of the first and second and
The last is three times the second?

Answer: 1349.

7. I am always hungry, I must always be fed.
The finger I lick will soon turn red.

Answer: fire.

8. A precious stone, as clear as diamond.
That shuns the sun’s bright fire.
Though you can walk on water with its power,
Try to keep it, and it’ll vanish ere an hour.

Answer: ice.

9. I am sometimes strong
And sometimes weak,
But I am nobody’s fool.
For there is no language that I can’t speak,
Though I never went to school.

Answer: An echo.

How did you do on these riddles, fellow puzzler? Let us know in the comments section below!

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Riddle Me This!

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[Image courtesy of Forbidden Planet.]

A PuzzleNationer reached out to me recently and asked about riddles. Specifically, he was asking about wordplay riddles, the ones that can take you a bit of time to mentally unravel, rather than the jokey riddles found in most children’s puzzle books.

You see, he’s a Dungeon Master, the man runs a Dungeons & Dragons game, shaping the story for the other players. He was about to lead his players into a lost catacomb left behind by a crafty wizard known for his love of wordplay, and he needed ideas for riddles that might challenge his players.

Thankfully, I had the perfect resource for him.

[No, not that guy… (Image courtesy of Nyafuu Archive.)]

The puzzly crew at Raging Swan Press foresaw the need for something like this, and years ago, they assembled three handbooks about riddles for anyone who might need them. This series is called “So What’s the Riddle Like, Anyway?” Downloadable PDFs of the books can be found on DriveThruRPG.com.

From the book’s introduction, where the authors present possible scenarios:

Your PCs are deep in the dungeon and have just encountered a terrifying monster which they have no chance of defeating. Luckily, the monster is bored and challenges the party to a riddling contest instead of simply just killing them. Alternatively, the party have encountered a sentient statue that will not let them past until they have answered three riddles correctly.

I am a huge fan of Raging Swan, because they’re all about providing additional content for roleplaying games in order to make the games more varied and interesting, and they price these expansions and idea-boosters very affordably.

For instance, each of the three editions of “So What’s the Riddle Like, Anyway?” are only $1.99 apiece.

And I figured, why not pit the puzzly minds of the PuzzleNation readership against the crafty campaign creators of Raging Swan Press.

Enjoy!

Volume I of the series not only walks the reader through the process of designing and choosing riddles for your game, but also instructs you on how best to use the riddles to advance your story. Volume I also offers some examples to get you started.

1. What always runs but never walks,
Often murmurs, never talks,
has a bed but never sleeps,
Has a mouth but never eats?

2. What has a head and a tail, but no body?

3. I can be cracked, I can be made.
I can be told, I can be played.

Volume II delves deeper into the puzzlier aspect of riddles, employing pattern identification, word problems, and brain teasers to offer another possible challenge for your players.

4. What should the tenth number in this series be? 3, 3, 5, 4, 4, 3, 5, 5, 4

5. A carpenter was in a terrible hurry. He had to work as quickly as possible to cut a very heavy ten‐foot plank into ten equal sections. If it takes one minute per cut, how long will it take him to get the ten equal pieces?

6. Can you find a four‐digit number in which:
The first digit is one‐third the second digit,
The third is the sum of the first and second and
The last is three times the second?

Volume III rounds out the trilogy with numerous traditional riddles about various aspects of the standard medieval roleplaying setting. Riddles about elements, dragons, weapons, creatures, and more await you inside this slim tome.

7. I am always hungry, I must always be fed.
The finger I lick will soon turn red.

8. A precious stone, as clear as diamond.
That shuns the sun’s bright fire.
Though you can walk on water with its power,
Try to keep it, and it’ll vanish ere an hour.

9. I am sometimes strong
And sometimes weak,
But I am nobody’s fool.
For there is no language that I can’t speak,
Though I never went to school.

How did you do on these riddles, fellow puzzler? Let us know in the comments section below!

Thanks for visiting PuzzleNation Blog today! Be sure to sign up for our newsletter to stay up-to-date on everything PuzzleNation!

You can also share your pictures with us on Instagram, friend us on Facebook, check us out on Twitter, Pinterest, and Tumblr, and explore the always-expanding library of PuzzleNation apps and games on our website!