# Two Brain Teasers, Courtesy of Conway

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

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.

[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.”

[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!)

[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:

[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.

[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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# Puzzly Ideas to Keep You Busy!

We’re all doing our best to keep ourselves and our loved ones engaged, entertained, and sane during these stressful times.

And after weeks of doing so, it’s possible you’re running out of ideas.

But worry not! Your puzzly pals at PuzzleNation are here with some suggestions.

Please feel free to sample from this list of activities, which is a mix of brain teasers to solve, puzzly projects to embark upon, treasure hunts, unsolved mysteries, ridiculous notions, creative endeavors, and a dash of shameless self-promotion.

Enjoy, won’t you?

Puzzly Ways To Get Through Self-Quarantine

In all seriousness, we hope these ideas help you and yours in some small way to make the time pass in a fun and puzzly fashion. Be well, stay safe, and happy puzzling.

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# A Coin Puzzle: My Two Cents (Plus 97 More)

Our friends at Penny Dell Puzzles recently shared the following brain teaser on their social media:

Naturally, we accepted the challenge.

Now, before we get started with this one, we have to add one detail: which coins we’re allowed to use. It’s safe to assume that pennies, nickels, dimes, and quarters are available, but the question doesn’t say anything about half-dollar coins.

So we’re going to figure out the correct answer without half-dollar coins available, and then with half-dollar coins available.

Let’s begin.

[Image courtesy of How Stuff Works.]

The easiest way to get started is to figure out the smallest number of coins we need to make 99 cents, since that’s the highest number we need to be able to form. Once we have that info, we can work backwards and make sure all the other numbers are covered.

For 99 cents, you need 3 quarters, 2 dimes, and 4 pennies. That’s 25 + 25 + 25 + 10 + 10 + 1 + 1 + 1 + 1 = 99.

Right away, we know we’re close with these 9 coins.

You don’t need more than 3 quarters, for instance, because your possible totals are all below \$1.

Now, let’s make sure we can form the numbers 1 through 24 with our chosen coins. (If we can, we’re done, because once we’ve covered 1 through 24, we can simply add one quarter or two quarters to cover 25 through 99.)

Our four pennies cover us for 1 through 4. But wait, there’s 5. And we can’t make 5 cents change with 4 pennies or 2 dimes. In fact, we can’t make 5, 6, 7, 8, or 9 cents change without a nickel.

So let’s add a nickel to our current coin count. That makes 3 quarters, 2 dimes, 1 nickel, and 4 pennies. (Why just 1 nickel? Well, we don’t need two, because that’s covered by a single dime.)

Our four pennies cover 1 through 4. Our nickel and four pennies cover 5 through 9. Our dime, nickel, and four pennies cover 1 through 19. And our two dimes, one nickel, and four pennies cover 1 through 29. (But, again, we only need them to cover 1 through 24, because at that point, our quarters become useful.)

That’s all 99 possibilities — 1 through 99 — covered by just ten coins.

[Image courtesy of Wikipedia.]

Well, we can apply the same thinking to a coin count with a half-dollar. For 99 cents, you need 1 half-dollar, 1 quarter, 2 dimes, and 4 pennies. That’s 50 + 25 + 10 + 10 + 1 + 1 + 1 + 1 = 99.

Now, we make sure we can form the numbers 1 through 49 with our chosen coins. (Once we can, we can simply add the half-dollar to cover 50 through 99.)

Once again, we quickly discover we need that single nickel to fill in the gaps.

Our four pennies cover 1 through 4. Our nickel and four pennies cover 5 through 9. Our dime, nickel, and four pennies cover 1 through 19. Our two dimes, one nickel, and four pennies cover 1 through 29. And our one quarter, two dimes, one nickel, and four pennies cover 1 through 54. (But, again, we only need them to cover 1 through 49, because at that point, our half-dollar becomes useful.)

That’s all 99 possibilities — 1 through 99 — covered by just nine coins.

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# Brain Teaser Week: Answers Edition!

Did you enjoy Brain Teaser Week, fellow puzzlers and PuzzleNationers? We certainly hope so! It was a fun experiment in dedicating an entire week to a particular type of puzzle.

We gave you three puzzles to challenge your deductive, mathematical, and puzzly skills, and now it’s time to break them down and explain them.

Tuesday’s Puzzle:

A set of football games is to be organized in a “round-robin” fashion, i.e., every participating team plays a match against every other team once and only once.

If 105 matches in total are played, how many teams participated?

If every team plays every other team once, you can easily begin charting the matches and keeping count. With 2 teams (Team A and Team B), there’s 1 match: AB. With 3 teams (A, B, and C), there are 3 matches: AB, AC, BC. With 4 teams (A, B, C, and D), there are 6 matches: AB, AC, AD, BC, BD, CD. With 5 teams (A, B, C, D, and E), there are 10 matches: AB, AC, AD, AE, BC, BD, BE, CD, CE, DE.

Now, we could continue onward, writing out all the matches until we reach 105, but if you notice, a pattern is forming. With every team added, the number of potential matches increases by one.

With one team, 0 matches. With two teams, 1 match. With three teams, 2 more matches (making 3). With four teams, 3 more matches (making 6). With five teams, 4 more matches (making 10).

So, following that pattern, 6 teams gives us 15, 7 teams gives us 21, and so on. A little simple addition tells us that 15 teams equals 105 matches.

Thursday’s Puzzle:

You want to send a valuable object to a friend securely. You have a box which can be fitted with multiple locks, and you have several locks and their corresponding keys. However, your friend does not have any keys to your locks, and if you send a key in an unlocked box, the key could be copied en route.

How can you and your friend send the object securely?

(Here’s the simplest answer we could come up with. You may very well have come up with alternatives.)

The trick is to remember that you’re not the only one who can put locks on this box.

Put the valuable object into the box, secure it with one of your locks, and send the box to your friend.

Next, have your friend attach one of his own locks and return it. When you receive it again, remove your lock and send it back. Now your friend can unlock his own lock and retrieve the object.

Voila!

Friday’s Puzzle:

The owner of a winery recently passed away. In his will, he left 21 barrels to his three sons. Seven of them are filled with wine, seven are half full, and seven are empty.

However, the wine and barrels must be split so that each son has the same number of full barrels, the same number of half-full barrels, and the same number of empty barrels.

Note that there are no measuring devices handy. How can the barrels and wine be evenly divided?

For starters, you know your end goal here: You need each set of barrels to be evenly divisible by 3 for everything to work out. And you have 21 barrels, which is divisible by 3. So you just need to move the wine around so make a pattern where each grouping (full, half-full, and empty) is also divisible by 3.

• 7 full barrels
• 7 half-full barrels
• 7 empty barrels

Pour one of the half-full barrels into another half-full barrel. That gives you:

• 8 full barrels
• 5 half-full barrels
• 8 empty barrels

If you notice, the full and empty barrels increase by one as the half-full barrels decrease by two. (Naturally, the total number of barrels doesn’t change.)

So let’s do it again. Pour one of the half-full barrels into another half-full barrel. That gives you:

• 9 full barrels
• 3 half-full barrels
• 9 empty barrels

And each of those numbers is divisible by 3! Now, each son gets three full barrels, one half-full barrel, and three empty barrels.

How did you do, fellow puzzlers? Did you enjoy Brain Teaser Week? If you did, let us know and we’ll try again with another puzzle genre!

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# Brain Teaser: A Curious Way to Tell Time

We love tackling brain teasers, riddles, math puzzles, and mind ticklers here at PuzzleNation Blog.

Our friends at Penny Dell Puzzles also enjoy putting their puzzly skills to work by posting various brain teasers..

And one of our mutual readers was hoping we could explain how to solve a puzzle that recently made the rounds on Penny’s social media accounts.

We’d be happy to!

Today, let’s take a look at a brain teaser all about time.

So, where do we begin?

Let’s start by breaking that sentence down to make it easier to parse: “The number of hours left today is half of the number of hours already passed.”

Well, a simpler way of saying that is “the number of hours already passed is twice the number of hours left.”

So if you have the number of hours left — let’s call that X — then the number of hours already passed is twice X, or 2X. Between X and 2X, that’s your entire day covered.

Sorting that out gives us the simple formula of X + 2X = 24, since there are 24 hours in a day.

That easily becomes 3X = 24, and simple division tells us that X = 8.

So X, the number of hours left today, is 8. Which means that twice X, or 16, is the number of hours already passed.

And if there are 8 hours left in the day (or 16 hours already passed), that means it’s 4 PM.

Most of the time, brain teasers are all about efficiently organizing the information we have.

That allows us to figure out how best to use that information to move forward and solve the puzzle. This is just as true with a relatively straightforward brain teaser like this as it is with a complicated logic puzzle with all sorts of pieces to put together.

It’s all about figuring out what we know, what we don’t, and how what we DO know can lead us to what we don’t.

That’s just part of the fun.

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