Tag Archives: math

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!)

JHC-GOL-600x170px

[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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Winning Monopoly With Math!

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I’m always on the hunt for tips to make myself a better puzzler and gamer. Sometimes you stumble across those tips in unexpected places.

For instance, I was reading, of all things, a book about mathematics and Christmas — The Indisputable Existence of Santa Claus: The Mathematics of Christmas — and inside, I found a statistical analysis of the best strategy for winning a game of Monopoly.

Yes, we’ve discussed this topic before, but even that previous deep-dive into the mechanics of the game wasn’t as thorough or as revealing as the work by Dr. Hannah Fry and Dr. Thomas Oleron Evans in this Christmas-fueled tome of facts and figures.

They started with a breakdown of how your first turn could go, based on dice rolls. This is the same breakdown as in our previous post, but with some important differences. For instance, they also considered the chances of going to jail after multiple doubles rolls.

Also, they covered the statistical impact of how landing on card spaces can affect where you land on your first turn. The Community Chest is a curveball, because of the possible sixteen cards, three will send you somewhere on the board: Go, Mediterranean Ave, and Jail.

A simple statistical analysis is complicated even further by the Chance cards — nearly half of the sixteen cards send you elsewhere: Go, Income Tax, St. Charles Place, Pennsylvania Railroad, Illinois Ave., Jail, and Boardwalk.

If you extrapolate forward from this point, you uncover some interesting patterns:

The orange property set benefits from all the ex-cons leaving their cells, and after their next turn the reformed criminals will likely end up somewhere between the reds and yellows… Illinois Avenue, with its own dedicated Chance card directing people to it, gets an extra boost, making it the second most visited square on the board.

…

The property that is visited least frequently is Park Place, where players spend just 2.1% of their time.

Check out this graph. This shows potential earnings from each complete color set, with the dotted line marking the point where your purchase of the property is canceled out by how much the property has earned in rent thus far. Everything above that is profit.

As you can see, blue and brown properties start close to the dotted line, because they’re affordable to buy and build on. The standouts on this graph are New York Avenue (which earns $30 a roll up through thirty rolls statistically) and Boardwalk, which is an expensive investment, but pays off handsomely down the line, remaining the top earning spot past thirty rolls.

Of course, that’s only single properties, and you can’t build on single properties. Let’s look at a chart for full color set revenue:

Some of our previous findings change radically. Boardwalk’s rating drops significantly, because of Park Place’s relative infrequency of being landed on (as we mentioned above).

So which properties should you nab to give yourself the best chance of winning? Well, that depends on how long the game lasts.

The average game of Monopoly takes approximately thirty turns per player, so the larger the number of players, the longer the game will last.

So, for a two-player game, your best bet is to go after the light blue or orange sets, since they’re better in the short term, and the odds are in your favor if the game stays short.

In a three- or four-player game, the orange and red sets are better, because the game is likely to last a while.

And if five or more people are playing, you’re really playing the long game, so the green set becomes your best chance for success.

What about building on those properties? Well, Fry and Evans considered that as well. If you’re playing against multiple opponents and know you’ll be in for a long game, then you definitely want to buy and place houses. But don’t fear if the first house takes a long time to start paying for itself.

As it turns out, your best strategy is to put three houses on your properties as quickly as possible, because the third house is the fastest to recoup on investment. So once the three houses are in place on each property, you can rest for a bit and regenerate your bank before investing further.

And there you have it. Better gaming through mathematics! The only thing better would be, well, playing practically any other game.

Kidding! (But not really.)

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Four Dimensional Hats: A Visual Wonder!

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[Image courtesy of Brilliant.org.]

The mobius strip is one of the simplest objects in the world, and yet, the most mind-bending. If you take a strip of paper, add a single twist, and tape the ends together, you transform a two-sided object into a single-sided object. It becomes one continuous surface.

(We’ve discussed the concept briefly in the blog before, but in bagel form.)

But did you know that you can take that idea a step further and end up with this?

[Image courtesy of math.union.edu.]

This is a Klein bottle, an object with one continuous surface. If you trace a path along the surface, you will traverse from the “inside” to the “outside” and back again without breaking stride.

Yes, the word “bottle” is a bit of a misnomer, since this won’t actually hold any liquids; they would just flow along the surface, going “inside” and back “out” without pooling anywhere. This is a result of a mistranslation, as the German word “flache” (surface) was translated as “flasche” (bottle).

This limerick sums up the Klein bottle nicely:

A German topologist named Klein
Thought the Mobius Loop was divine.
Said he, “If you glue
The edges of two,
You get a weird bottle like mine.”

[Image courtesy of Pinterest.]

Although the Klein bottle can’t quite exist as a three-dimensional object — since the object has to pass through itself, which can only happen in four dimensions — we can come close enough to create some impressive approximations, like the glass “Klein bottles” pictured above.

YouTuber and physics student Toby Hendy has even managed to create a technique to knit yourself a Klein bottle hat! Check it out:

Although it’s not an optical illusion, it’s still a visual puzzle for the eyes and the mind, one that has captured the imaginations of mathematicians, artists, and many others throughout the years.

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PuzzleNation Product Review: Math Fluxx

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math-fluxx

[Note: I received a free copy of this game in exchange for a fair, unbiased review. Due diligence, full disclosure, and all that.]

Fluxx has been one of Looney Labs‘s flagship products for over two decades now. It’s the card game with constantly changing rules, a game where the path to victory can vanish or appear at a moment’s notice.

But after Batman Fluxx, Firefly Fluxx, Holiday Fluxx, and many other versions, what more can they do with the concept to keep it fresh and interesting?

As it turns out, plenty. And with their latest release, Math Fluxx, the Looney Labs crew proves they still have plenty of tricks up their sleeves.

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Now, anyone who has played Fluxx in the past is familiar with the basic gameplay: you collect keeper cards and put them into play. Different combinations of keeper cards complete different goals, and each player has the chance to put different keeper cards and goal cards into play in order to win.

Along the way, players affect how the game is played by utilizing action cards and new rule cards which alter what players can and can’t do. Suddenly, you’ll have to trade your hand with another player, or start drawing three cards each turn instead of one.

But instead of matching images like you do in most versions of Fluxx, in Math Fluxx players have to use keeper cards with numbers on them in order to complete different mathematical goals.

mathfluxx1

Some of the goals are simple, like having 4 and 2 as keepers to make 42 (the answer to life, the universe, and everything). But other goals are more complex, like forming two pairs of keeper cards like a poker hand, or having the highest score on the table in keeper cards.

For example, there’s a goal where you win if you’re displaying your own age with keeper cards. But since people playing will probably have different ages (and therefore, different keepers for that goal), you could lose by playing that goal too early.

Achieving these goals requires more strategy than your usual game of Fluxx — which is built more on seizing opportunities, since the gameplay is often quite chaotic — and the game’s creators doubled down on this by introducing new rule cards that let you achieve some of the goals in different ways.

For instance, instead of forming 42 with a 2 card and a 4 card, one new rule would allow you to complete that goal by playing keeper cards that, when multiplied, form 42.

These new wrinkles add a tremendous amount of depth to the gameplay (and I haven’t even mentioned the meta rule cards that alter gameplay for an entire session rather than a few turns, if players are feeling particularly ambitious).

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Math Fluxx also cleverly sneaks in real-world mathematical concepts for younger players, in case you’re looking for a stealthy way to reinforce learning through playing games.

I was thoroughly impressed by the variety in new rules, goals, and gameplay tweaks introduced by Math Fluxx. It shows that there’s plenty of life in the Fluxx franchise, and that spirit of innovation and playfulness infuses each round of play, encouraging players to be just as clever and creative with their own gameplay.

Math Fluxx will be available March 9th, but you can preorder it by clicking here! And to check out all of our reviews of Looney Labs games and products, click here!

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A puzzle in your pocket

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Brain teasers come in all shapes and sizes, but there’s one particular brand of brain teaser that fits in your pocket. Today we’re talking about matchstick puzzles (or toothpick puzzles).

Matches as we know them (relying on friction to ignite, rather than dipping or crushing) were invented in 1826 by English chemist John Walker, and in the decades that followed, numerous improvements were made, especially in terms of safety and ease of use. Matchsticks soon grew ubiquitous and match companies started putting little puzzles on their boxes.

And the matchstick puzzle was born.

(I have no historical documentation to back me up on this, but I suspect that bar bets also played a role in the rise of matchstick puzzles, because the sort of cleverness and trickery that goes into solving some of these puzzles would be perfectly at home in the repertoire of someone looking to con a few free drinks out of fellow tipplers.)

So, for the uninitiated, what are matchstick puzzles?

These are rearrangement or transformation puzzles, where you’re given a certain shape (laid out in matchsticks, toothpicks, straws, pencils, or anything else of equal length), and you have to move the items into another shape or configuration. Sometimes, it’s simply about placing the matchsticks economically, but other times, you have to get crafty and think outside the box to complete your task.

For instance, here’s the first matchstick puzzle I ever remember seeing:

[This image, and the one below, courtesy of Matchstick Puzzles on Blogspot.]

You have two triangles formed from six matchsticks. Move one matchstick to make four triangles.

Now, you could easily use all of these matchsticks to make four triangles, but that would involve moving more than one of them. So clearly there’s something else at work here if you only need to move one to solve the puzzle.

That something, in this case, is a little visual trickery.

As you can see, you turn one triangle into a numeral four, making the matchsticks literally read out “4 triangle.” Sneaky sneaky.

There are literally hundreds of these puzzles if you go hunting for them. (I found a treasure trove of them here.)

A curious variation, though, applies the same rules to mathematical formulas laid out in matchstick form.

Here’s one that’s been making the rounds on Facebook recently:

Now, the big difference between these mathematical ones and the shape ones mentioned above, as far as I’ve found, is that the math ones are far more alternate prone.

For instance, this equation puzzle has at least four solutions that I’ve found:

  • You can move one match to make the 6 a 0, so that 0+4=4.
  • You can move one match from the 6 to the second 4 to make the 6 a 5 and the 4 a 9, so that 5+4=9.
  • You can move one match from the plus sign to the 6 to make the plus sign a minus sign and the 6 an 8, so that 8-4=4.
  • You can move one match from the plus sign to the equal sign to make the plus sign a minus sign and the equal sign a doesn’t-equal sign, so that 6-4 does not equal 4.

As you can see, with matchstick puzzles,  the possibilities are endless and the building blocks — whether matches, toothpicks, Q-Tips, or straws — are easily accessible.

I’ll leave you one more to ponder, this time provided by the folks at IO9:

Using six matchsticks of equal length, create four identical, equilateral triangles. There’s no need for snapping, burning, or otherwise altering the matchsticks.

Good luck!

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Keep your Get Out of Jail Free card handy.

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Recently, friend of the blog Chris Begley sent me an article about some interesting math facts. Most of them were about probability and how many people misinterpret the likelihood of various events happening based on bad assumptions about probability.

For instance, since you have a 50/50 chance of heads or tails when you flip a coin, it seems logical that if you flipped a coin ten times, you’d get heads five times and tails five times, whereas in reality, it’s common to have runs of one result or the other that fly in the face of that simple 50/50 assumption.

But that wasn’t the fact that caught my eye. I thought it was much more intriguing that not all spaces on a Monopoly board have an equal likelihood of being landed on.

And that can affect how you play. For instance, if you believe each spot has an equal chance of being landed on — 1 in 40, given the 40 squares on the board — you might opt to buy all three colors in a given area to give yourself a 3/40 chance (7.5%), or you might go for all 4 railroads to give yourself a 4/40 chance (10%).

[A breakdown of spaces and likelihood of landing, based on the UK version.
(Chance and Community Chest cards differ between UK and US versions,
though probabilities for spaces in the US version are quite similar.)]

But that’s not how Monopoly actually works. Some spaces are far more likely than others. This is partly due to rolling two dice every time you move (which makes 6, 7, or 8 spaces the most likely results). There are also rule cards that make some squares more likely than others.

The most common space to land on is Jail (due in no small part to the Go to Jail square and where Chance and Community Chest cards send you). The most common PROPERTY to land on is Illinois Avenue, followed by B&O Railroad, Tennessee Avenue, New York Avenue, and Reading Railroad.

[A breakdown of space probabilities for the US version of the game.]

On the flip side, Mediterranean Avenue is the least likely to be landed on, followed by Baltic Avenue, Luxury Tax, Park Place, and Oriental Avenue. (Again, the Go to Jail square comes into play, as Park Place is seven squares away and the most common dice roll is 7.)

I like that a little properly applied math might make you a better Monopoly player. (Though if I’m going to walk the Boardwalk, I’d rather be playing The Doom That Came to Atlantic City.)

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