Monthly Archives: June 2020

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

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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 #1, The 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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Intersections of Puzzle and Poetry

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The more you look, the more you can find puzzles in all sorts of interesting places. We find them in literature, in historical documents, and in popular culture.

So it should come as no surprise that puzzles can be found in the world of poetry as well.

We’ve covered a few examples where poetry and puzzles have overlapped in the past, whether it’s the creations of Peter Valentine, the works of Edgar Allan Poe, or the art of carmina figurata.

carminafig7

But that’s only scratching the surface.

One of the most common ways that puzzly techniques find their way into poetry is through acrostics. Acrostics spell out messages with the first letter of each line or verse.

One of the most famous is a poem by Lewis Carroll at the end of Through the Looking-Glass where he reveals the identity of the girl who inspired his famous stories:

A boat beneath a sunny sky,
Lingering onward dreamily
In an evening of July—

Children three that nestle near,
Eager eye and willing ear,
Pleased a simple tale to hear—

Long has paled that sunny sky:
Echoes fade and memories die.
Autumn frosts have slain July.

Still she haunts me, phantomwise,
Alice moving under skies
Never seen by waking eyes.

Children yet, the tale to hear,
Eager eye and willing ear,
Lovingly shall nestle near.

In a Wonderland they lie,
Dreaming as the days go by,
Dreaming as the summers die:

Ever drifting down the stream—
Lingering in the golden gleam—
Life, what is it but a dream?

Carroll certainly offers the most famous example, but I must confess that my favorite example comes from a story on Wikipedia. Poet Rolfe Humphries was banned from Poetry Magazine for life for an acrostic aimed at a diplomat and former president of Columbia University. The acrostic quite bluntly read “Nicholas Murray Butler is a horse’s ass.”

Of course, the message reading down — also known as an acrostich — isn’t the only way these messages can be hidden.

There are also examples of mesostich — where the word or message is spelled with letters in the middle of the verse — and telestich, where the last letters of each line spell a name or message.

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[Image courtesy of Twitter.]

These techniques were also used in ancient Greek inscriptions, where one particular example, AL205, featured acrostich, mesostich, and telestich messages at the same time.

Other puzzly stylings have also allowed poets to flex their wordplay muscles.

For instance, David Shulman wrote a 14-line sonnet about George Washington’s famous river crossing where every line is an anagram of “Washington crossing the Delaware”:

A hard, howling, tossing water scene.
Strong tide was washing hero clean.
“How cold!” Weather stings as in anger.
O Silent night shows war ace danger!

The cold waters swashing on in rage.
Redcoats warn slow his hint engage.
When star general’s action wish’d “Go!”
He saw his ragged continentals row.

Ah, he stands – sailor crew went going.
And so this general watches rowing.
He hastens – winter again grows cold.
A wet crew gain Hessian stronghold.

George can’t lose war with’s hands in;
He’s astern – so go alight, crew, and win!

There are also ABC poems, a form where the goal of each poem is to use words starting with each letter of the alphabet in order. You can find some entertaining and impressive examples here.

Some poets, however, have flipped the puzzle poem on its head by treating the poems like puzzles. The folks at UVA’s Puzzle Poetry group utilize Tetris-like puzzle pieces with words on them to assemble poems.

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[Image courtesy of the University of Virginia.]

The concept dates back to 2017, a creation of Neal Curtis and Brad Pasanek, serving as a way to both explore and deconstruct the art of poetry itself by making a puzzle out of it.

It’s a very cool idea, reminiscent of how magnetic poetry sets allow you to turn your fridge into a canvas by assembling and reworking the order of the various available words.

Puzzles by their very nature are about finding a solution, bringing order out of chaos, whether it’s assembling puzzle pieces, answering devious crossword clues to fill a grid, or unraveling a tricky brain teaser that pushes you to think in a different way.

And since poetry is all about expressing truths in a personal way, it makes a lovely sort of sense that puzzly techniques would intertwine with this thoughtful, elusive form of art.

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