Tag Archives: mathematics

The Oldest Puzzle in History?

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archimedes

[Archimedes, looking disappointed for some reason. Image courtesy of Wikipedia.]

Imagine the first puzzle. The very first one. What form would it take? Would it involve words? Numbers? Pictures? Would it be a riddle? A jigsaw? Would there be pieces to move around and place? Would it require scratchings in ink and quill to solve, or marks on a clay tablet?

It’s hard to visualize, isn’t it?

The subject of today’s blog post was probably not the first puzzle, but it’s the oldest puzzle that we can trace back to its origins. And those origins are more than two thousand years in the past.

Fellow puzzlers, allow me to introduce the Ostomachion.

ostomachion2

[The puzzle can be found in paper, wood, plastic, and other forms. The original was supposedly made from bone. Image courtesy of Oh So Souvenir.]

The Ostomachion, also known as the Stomachion, the Syntemachion, the Loculus of Archimedes, or Archimedes’ Square, consists of 14 shapes that can be arranged to fill a square.

Created by Archimedes in the 3rd century B.C., the Ostomachion might’ve vanished from history if not for the clever investigative skills of researchers. You see, the Ostomachion was among other writings by Archimedes that were transcribed into a manuscript in 10th-century Constantinople. The manuscript was then scraped clean and reused in the 13th-century as a Christian religious text (becoming a palimpsest in the process), where it remained until at least the 16th century.

archimedespalimpsest

[Image courtesy of Harvard. Yes, that Harvard.]

Thankfully, the erasure was incomplete, and in 1840, a Biblical scholar named Constantin von Tischendorf noted the Greek mathematics still visible beneath the prayer text. Another scholar recognized it as the work of Archimedes.

After changing hands multiple times, being sold (most likely illegally), modified by a forger, and then finally allowed to be scanned with UV, infrared, and other spectral bands, revealing the full mathematical text (as well as other works, all of which are now available online).

This palimpsest is the only known copy of both the Ostomachion and another Archimedean work, “The Method of Mechanical Theorems.”

ostoshapes

[Shapes to be solved. Image courtesy of Latinata.]

So, all that trouble for a place-the-pieces puzzle? Obviously there’s a bit more at play here.

After a solver has managed to fill the square , they are invited to use the pieces to make a variety of different shapes (similar to tangram puzzles). Players could compete to see who could use all of the pieces to form the different shapes first. It’s believed that this is where the name Ostomachion came from, as it translates to “bone fight” in Greek.

But, naturally, Archimedes didn’t stop there, delving into the mathematics of the puzzle itself, and trying to calculate how many unique solutions there were to the Ostomachion square. How many different ways could you fill the square?

osto-loculus17

[Cutler’s 17th solution. Image courtesy of MathPuzzle.com.]

That question wouldn’t be answered until 2003, when Bill Cutler — a mathematician with a doctorate in mathematics from Cornell — and some brute-force computing figured out that there were 17,152 solutions.

Seventeen thousand.

But, wait, it’s a square. So, technically, there must be quite a bit of overlap in those solutions, since some of them would be rotations or reflections of other solutions.

So what’s the real answer?

536. 536 distinct solutions. (You can view them all here.)

And it only took 2200 years to find out.

That, my fellow PuzzleNationers, is quite a puzzle.

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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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Let’s Make a Deal!

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It’s a scenario every game show fan knows well. You’ve got three doors to choose from, and one of those doors will open to reveal a fabulous prize.

After you’ve made your choice (let’s say Door #2), our affable host Monty Hall plays Devil’s Advocate by opening one of the doors you didn’t choose (let’s say Door #1), revealing a goat or other lackluster result.

And then, Monty offers you a chance to change your mind. Will you stick with the door you initially chose, or will you switch to the other unopened door (Door #3)?

The average player sees two choices, Door #2 and Door #3, which on the surface sounds like a 50/50 shot, a coin flip. So would it surprise you to learn that people who switched from one door to the other doubled their chances to win the fabulous prize?

This is known as the Monty Hall Problem, an example of how statistics aren’t always what they seem, and it has puzzled people for decades.

It’s counterintuitive, isn’t it? I mean, you have two choices, so the odds should be 50/50. But you’re forgetting that third door that Monty eliminated. That third door makes all the difference, statistically speaking.

Let’s break it down. Your initial choice is between 3 doors, meaning you have a 1 in 3 chance of picking the correct door, and a 2 in 3 chance of picking the wrong one.

When Monty opens that other door, the odds haven’t changed. Only the number of options available has changed. Your door is still a 1 in 3 chance of being correct and a 2 in 3 chance of being wrong. But the remaining door now has a 2 in 3 chance of being correct!

So what appeared to be a coin flip between sticking with your choice and switching is now heavily weighted toward switching!

There have been several real-world tests of the Monty Hall Problem, and all of them have consistently shown that the people who switch were twice as likely to open the winning door!

The real puzzle here is how we fool ourselves. We take the numbers at face value — 3 doors become 2 doors, so a 1 in 3 chance becomes a 1 in 2 chance — and actually hurt our chances with those seemingly simple assumptions.

Being able to reconsider your assumptions is a major tool in the puzzler’s solving kit. Plenty of tricky crossword clues depend on you associating the clue with one thing, when the answer is something quite different.

After all, if you saw the clue “Unlocked” for a four-letter entry, you’d probably try OPEN before you tried BALD. Clever constructors are counting on that.

So be sure to remember Monty Hall and his three-door conundrum the next time you’re stumped on a puzzle. Maybe the answer is as simple as trying another door.

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