Answer to the Fiendish Second Conway Puzzle, The Ten Divisibilities!

John_H_Conway_2005_(cropped)

Last month, in honor of mathematician and puzzly spirit John Horton Conway, we shared two of his favorite brain teasers and challenged our fellow PuzzleNationers to crack them.

Two weeks ago, we shared the solution to puzzle #1The Miracle Builders, and offered a few hints for puzzle #2, The Ten Divisibilities.

Now that we’ve heard from a few solvers who either conquered or got very close to conquering the second puzzle, we happily share both the solution and how we got there.


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…]

And here are the hints we offered to help:

-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.


The solution is 3816547290.

So, how do we get there?

First, we use process of elimination.

Any number divisible by 10 must end in a zero, so j = 0.

Any number divisible by 5 must end in a zero or a five, so e = 5 (because each digit only appears once).

That gives us abcd5fghi0.

But that’s not all we know.

If a number is divisible by an even number, that number must itself be even. So that means b, d, f, and h must all be even numbers (i.e. some combination of 2, 4, 6, and 8). That also means that a, c, g, and i must all be some combination of the remaining odd numbers (1, 3, 7, and 9).

That’s a lot of information that will come in handy as we solve.

So, where to next? Let’s look at one of those even-numbered spots.

We’ve been told that abcd is divisible by 4. But any number is divisible by 4 if the last two digits are divisible by 4. So that means cd is divisible by 4.

So, if c is odd, d is even, and cd is divisible by 4, that limits the possibilities somewhat. cd must be 12, 16, 32, 36, 72, 76, 92, or 96.

So d is either 2 or 6.

That will be helpful in figuring out def. And knowing def is the key to this entire puzzle.


One of the clues we offered in our last post was that if the sum of a number’s digits is divisible by 3, then that number is also divisible by three. We know abc is divisible by 3, so that means a + b + c is also divisible by 3.

And if something is divisible by 6, then it’s also divisible by 3, so a + b + c + d + e + f is divisible by 3.

Here’s where things get a little tricky. Since a + b + c + d + e + f is divisible by 3, and a + b + c is divisible by 3, then when you subtract a + b + c from a + b + c + d + e + f, the result, d + e + f would also be divisible by 3.

Why is that helpful? Because it means we can look at def instead of abcdef, and we know a lot about def right now.

d is either 2 or 6. e is 5. f is either 2, 4, 6, or 8. And the sum of d + e + f is divisible by 3.

So that gives us two possibilities to deal with, either 2 + 5 + f, where the sum is divisible by 3, or 6 + 5 + f, where the sum is divisible by 3.

Since each number is only used once, that’s six possible equations:

  • 2 + 5 + 4 = 11
  • 2 + 5 + 6 = 13
  • 2 + 5 + 8 = 15
  • 6 + 5 + 2 = 13
  • 6 + 5 + 4 = 15
  • 6 + 5 + 8 = 19

Only 258 and 654 have sums divisible by 3, so they’re our two possibilities for def.

We’ll have to try both of them to see which is the correct choice. How do we do that?

Let’s start with the assumption that def is 258.


That would mean our answer is abc258ghi0. We know b and h have to be even numbers, and only 4 and 6 are left as options. Since fewer numbers are divisible by 8 than by 2, let’s look at abc258gh.

One of the other hints we offered was that if the last three digits of a number are divisible by 8, then the whole number is divisible by 8.

So that means if abc258gh is divisible by 8, then 8gh is divisible by 8. That’s much more manageable.

So, f is 8, h is 4 or 6, and g is either 1, 3, 7, or 9. That gives us eight possibilities for 8gh: 814, 834, 874, 894, 816, 836, 876, and 896.

Dividing each of these by 8 reveals only two possible choices: 816 and 896. That means, in this scenario, h is 6, b is 4, and our number is a4c258g6i0.

What’s next? Well, remember that trick we did with abcdef before? We’re going to do it again with abcdefghi.

Any number divisible by 9 is divisible by 3. Our rule of sums tells us that a + b + c + d + e + f + g + h + i is also divisible by 3. And since a + b + c + d + e + f is divisible by 3, subtracting it means that g + h + i is also divisible by 3.

With 816 and 896 as our possibilities for fgh, that means our possibilities for ghi are 16i and 96i. That gives us the following possibilities: 163, 167, 169, 961, 963, 967, where the sum of our answer must be divisible by 3.

  • 1 + 6 + 3 = 10
  • 1 + 6 + 7 = 14
  • 1 + 6 + 9 = 16
  • 9 + 6 + 1 = 16
  • 9 + 6 + 3 = 18
  • 9 + 6 + 7 = 22

963 is the only one that works, which gives us a4c2589630. With only 1 and 7 remaining as options, our possible solution is either 1472589630 or 7412589630.

But, if you divide either 1472589 or 7412589 by 7 — which is faster than running every one of the 10 conditions through a calculator — neither divides cleanly. That means 258 is incorrect.


I know that was a lot of work just to eliminate one possibility, but it was worth it. It means 654 is correct, so our solution so far reads abc654ghi0.

And we can use the same techniques we just employed with 258 to find the actual answer.

We know b and h have to be even numbers, and only 2 and 8 are left as options. Again, since fewer numbers are divisible by 8 than by 2, let’s look at abc654gh.

4gh is divisible is 8. So, f is 4, h is 2 or 8, and g is either 1, 3, 7, or 9. That gives us eight possibilities for 4gh: 412, 432, 472, 492, 418, 438, 478, and 498.

Dividing each of these by 8 reveals only two possible choices: 432 and 472. That means b is 8, and our number is a8c654g2i0.

Now, let’s look at ghi.

With 432 and 472 as our possibilities for fgh, that means our possibilities for ghi are 32i and 72i. That gives us the following possibilities: 321, 327, 329, 721, 723, 729, where the sum of our answer must be divisible by 3.

  • 3 + 2 + 1 = 6
  • 3 + 2 + 7 = 12
  • 3 + 2 + 9 = 14
  • 7 + 2 + 1 = 10
  • 7 + 2 + 3 = 12
  • 7 + 2 + 9 = 18

Okay, that leaves us four possibilities for ghi: 321, 327, 723, and 729.

Stay with me, folks, we’re so close to the end!

Let’s look at our four possibilities:

  • a8c6543210 (79)
  • a8c6543270 (19)
  • a8c6547230 (19)
  • a8c6547290 (13)

Next to each number, I’ve placed the only digits missing in each scenario, two for each.

That means there are only 8 possible ways to arrange the remaining numbers:

  • 7896543210
  • 9876543210
  • 1896543270
  • 9816543270
  • 1896547230
  • 9816547230
  • 1836547290
  • 3816547290

So let’s do what we did last time, and divide each chain at the seventh number by 7.

  • 7896543 / 7
  • 9876543 / 7
  • 1896543 / 7
  • 9816543 / 7
  • 1896547 / 7
  • 9816547 / 7
  • 1836547 / 7
  • 3816547 / 7

Only one of the chains can be cleanly divided by 7, and it’s 3816547.

Which means the solution for abcdefghij is 3816547290.


I know this was a monster of a solve — it rivals our Brooklyn Nine-Nine seesaw puzzle solution in complexity — but it’s one that every one of our fellow PuzzleNationers are capable of puzzling out.

How did you do on this diabolical brain teaser, folks? 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)

John_H_Conway_2005_(cropped)

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

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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PuzzleNation Product Review: Minecraft Magnetic Travel Puzzle

tfmine1

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

Minecraft is one of the biggest indie video game success stories of the last twenty years. A simple block-style game about building things (and destroying things) is now a multimedia empire, complete with toys, LEGOs, and of course, video games across numerous platforms.

It was only a matter of time before it made the leap to puzzles, and as it turns out, the clever folks at ThinkFun were just the designers to bring Minecraft into a puzzlier world.

Minecraft Magnetic Travel Puzzle pits the player against devious deduction puzzles with elements of the Minecraft universe included. By using the clues provided on each challenge card, the player must arrange three swords, pickaxes, and pieces of armor (all different colors, making nine unique game pieces) on the 3×3 crafting table in a particular pattern.

tfmine2

Completing the grid is the only way to bypass the ender dragon (who is placing these challenge obstacles in your path) and continue onto the next world in your journey.

The instructions, puzzles, solutions, game board, and pieces are all contained within the single spiral-bound game book, making this one of ThinkFun’s most portable products yet. The magnetic pieces are fairly sturdy, as is the game board, so it will hold up nicely to the rigors of travel (and being stuffed into various carry-on bags).

tfmine3

The gameplay itself is all about interpreting the clues provided with each challenge card. Some clues offer hints on where to place pieces according to color, others according to shape. Additional clues center around a given piece’s location on the grid or in relation to another piece.

For instance, in Beginner Challenge #5 in the image below, the solver gets two hints: one about color and the other about the game pieces.

All three of the blue pieces will be placed along the diagonal, according to the first hint. And according to the second hint, a piece of armor will be in the upper right corner and a pickaxe will be in the middle square. Combining these two hints tells us where to place the blue armor and blue pickaxe. And since only one blue gamepiece is left, the blue sword goes in the lower left corner.

tfmine4

Similarly, the combination of the yellow square in the center of the top row in the first hint and the sword image in the center of the top row in the second hint tells us where to place the yellow sword. Once that’s in place, we look at the remaining sword image on the second hint and know where to place the gray sword.

The gray square in the upper left corner of the first hint and the pickaxe image in the upper left corner of the second hint point to where to play the gray pickaxe (and the yellow pickaxe by process of elimination).

tfmine5

With two game pieces left and one unoccupied yellow square in the first hint, the solver can easily complete this challenge, besting the ender dragon’s latest obstacle and moving forward.

Once you graduate from the Beginner and Intermediate difficulty levels, you’ll face a new wrinkle: negative clues. Negative clues are layouts that must be avoided, so instead of telling you where to place a piece, they tell you expressly where NOT to place a piece, ratcheting up the difficulty.

For instance, in Advanced Challenge #25, the negative hints tell us that a gray gamepiece can never be directly below and to the right of a blue gamepiece, or above and to the left of a yellow gamepiece.

tfmine6

These restrictions will prove to be valuable hints going forward, often telling a savvy solver more about the layout of the crafting table than the regular clues!

By gradually teaching deductive reasoning — slowly introducing new ways to provide information and eliminate possibilities — the solver quickly grasps a key component of strategy and planning: “If this, then that” thinking.

This sort of cause-and-effect observation allows a solver to hold several pieces of information in your head at once, eliminating red herrings and unhelpful possibilities until you’re left with one solution that fits all the requirements. (Just as every Sudoku puzzle is an exercise in deduction, so is every challenge card in Minecraft Magnetic Travel Puzzle.)

Fun for younger solvers and older alike, ThinkFun’s latest deduction puzzle game turns Minecraft into Mindcraft, adding a valuable puzzly tool to the arsenal of every solver.

Minecraft Magnetic Travel Puzzle is available from ThinkFun and certain online retailers.


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Answers to Our Thanksgiving Logic Puzzle

We’re halfway to Christmas already, but remember Thanksgiving? It wasn’t all that long ago, we swear!

We celebrated the holiday by posting a fun little logic puzzle for our fellow puzzlers and PuzzleNationers. And we’re overdue to post the solution to our little dinner dilemma!

So, without further ado, let’s get to work!


Four girls — Emma, Taylor, Madison, and Morgan — are waiting for the Thanksgiving dinner.

Each is a different age (8, 9, 10, or 11) and looking forward to eating a different Thanksgiving staple (turkey, mashed potatoes, stuffing, or pumpkin pie).

Can you puzzle out the age and favorite Thanksgiving food of each girl from the clues below?

  • Madison is looking forward to eating turkey.
  • The girl who likes pumpkin pie is one year younger than Morgan.
  • Emma is younger than the girl that loves turkey.
  • The girl who likes stuffing is two years older than Morgan.

So where do we start? Simple. We start with Morgan.

Why Morgan? Because we have two clues connected to her that mention ages, clues 2 and 4.

There is a girl one year younger than Morgan and a girl two years older. Since the ages are 8, 9, 10, and 11, that means Morgan must be 9 years old.

thankspuzzle1

Now let’s look at clue 3. The girl who loves turkey is either age 9 or 10, since we already have foods for the other ages. Emma is younger than that girl. The only way Emma can be younger than age 9 or 10 is to be 8 years old.

So let’s fill that part in.

thankspuzzle2

Sticking with turkey-related clues, we can now look at clue 1. If Madison is the one who likes turkey, she has to be age 10, because that’s the only age in the chart with both the girl’s name and the favorite food uncompleted.

And by process of elimination, that means Morgan likes mashed potatoes and Taylor likes stuffing.

thankspuzzle3

How did you do, fellow puzzlers? Did you solve the logic puzzle in time for turkey dinner? Let us know in the comments section below! We’d love to hear from you.


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PuzzleNation Product Review: Puzzle Books Galore!

As part of our Holiday Puzzly Gift Guide, we reached out to companies far and wide to explore as big a swathe of the puzzle/game world as we could. And a plethora of puzzle books arrived in response.

With eight in total to cover in this review, we’re going to work from simplest to toughest in terms of difficulty, whilst bundling some books with similar puzzles or styles of presentation together for ease of navigation.

So please enjoy as we peruse offerings from USA Today, the Puzzle Society, and Andrews McMeel Publishing.


ampws

We start our puzzle book journey with objectively the easiest type of puzzle in the group: word searches.

Posh Simple Word Search collects grids and lists of hidden words to test your word recognition skills. The different sizes, themes for puzzles, and variations of word search puzzles (like an Eiffel Tower-shaped grid!) across more than 100 puzzles will have you looping words to your heart’s content.

Factor in a spiral binding that allows you to lay each page flat as you solve, and you’ve got a perfect intro to puzzles.

amppocketxwd1      amppocketxwd2

From word searches to another iconic and traditional puzzle type: crosswords.

For a travel-friendly puzzle book with eye-catching cover designs and enjoyable pocket-sized puzzles, look no further than Pocket Posh New Crosswords 1 and New Crosswords 2.

With fun, accessible clues and grids designed to test newer, less experienced solvers, Pocket Posh New Crosswords won’t stand in the way of a New York Times-level solver, but they will serve as a satisfying puzzle experience for solvers working their way up the difficulty ladder.

Featuring more than 50 puzzles each, these books are loaded with content created by The Puzzle Society’s pool of talented constructors. (All of whom are credited by name!)

ampxwd

For a step-up in difficulty and notoriety, check out USA Today’s Crossword Super Challenge.

Packed with 200 puzzles previously published in USA Today, this collection offers a range of difficulty levels depending on the constructor. And the names here are top-notch. Puzzly elites like Elizabeth Gorski, Martin Ashwood-Smith, Gail Grabowski, Frank Longo, and George Barany are featured in the collection, along with numerous contributions by USA Today‘s Crossword Editor, the inimitable Fred Piscop!

This array of 15×15 grids presents loads of different types of themed clues, serving as an ideal crash course in crosswords for solvers with a bit more experience but also have room to grow. Perfect for anyone who enjoys your local daily/weekly syndicated newspaper crossword.

It’s a little thick to make a great travel book — not as pocket-friendly as the Pocket Posh series — but it’s just right for an afternoon or two of cozy armchair solving.

ampsudoku

We then move from one world-conquering puzzle style to another that more recently took the world by storm: Sudoku.

Another in the USA Today series of Super Challenge titles, USA Today’s Sudoku Super Challenge is armed to the teeth with 200 Sudoku puzzles to challenge any fan of the infamous puzzle juggernaut.

Each puzzle is ranked on a scale of 1 to 5 stars in terms of difficulty, so you’ll be solving your way through increasingly tricky number puzzles the deeper you get into this book.

And despite being packed with hundreds of puzzles, this one will easily fit into a pocket, purse, or carry-on for any trip.

ampkurosu

Are you a Sudoku-savvy solver looking to test your number-placement skills in a new way? Posh Kurosu might just have what you’re seeking.

With dozens of examples of Kurosu puzzles — also known as noughts and crosses — this puzzle book packs a surprising amount of variety into a simple solving mechanic. Instead of nine digits to fill the grid, all you have are Xs and Os. And you can’t have more than two Xs or Os next to each other in any column, row, or diagonal.

This is the only kind of puzzle in this selection of puzzle books that I’d never encountered before, and it was a welcome change of pace to try my hand at something that felt familiar and yet fresh all at once. Posh Kurosu tests your logic and deduction chops in fun, unexpected ways.

amplogic

After collections of Sudoku and Kurosu puzzles, it feels appropriate to follow up with a puzzle book loaded with puzzles that test your logic and deduction skills in other ways.

USA Today’s Logic Super Challenge fits the bill nicely, mixing traditional story-driven logic problems (complete with those iconic solving grids to help you weed out false paths) with other logic-based puzzles like Killer Sudoku, Battleships, and Domino Search.

All of these puzzles will bend your brain around corners as you try to hold multiple facts in your head at the same time, waiting for them to fall into place and reveal a new piece of the overall puzzle solution.

And with 200 logic problems in various forms, you certainly won’t run out of devious deduction puzzles anytime soon.

mysteriousmansionsmall

But if you’re looking for a unique solving experience, something that is as visually immersive as it is engagingly puzzly, then you can’t go wrong with Daria Song’s The Mysterious Mansion.

Mixing lushly illustrated scenes with black and white drawings meant for you to color in, this narrative puzzle book incorporates mazes, spot-the-difference games, word searches, and other puzzly endeavors in a story about one girl’s journey through a strange and confusing mansion.

Designed to relax, engage, and puzzle the reader in equal measure, this book is one you could lose yourself in for hours. The gorgeous full-color illustrations are a feast for the eyes, and the puzzles are seamlessly woven into the art and story of each scene.

Daria Song gleefully takes activity books to the next level with this beautiful puzzle experience, a fairy tale that you not only help write, but make your own by doing so.


All of these puzzle books are available from Andrews McMeel Publishing as well as some local and online retailers. They’re also part of this year’s Holiday Puzzly Gift Guide!

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


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