Fix These Sixes!

[Image courtesy of Daily Brain Teaser.]

It’s a new year, and I’ve already got a new math puzzle for you. A friend discovered this one and sent it my way in the hopes that I’d be able to crack this diabolical brain teaser.

Fellow puzzlers and PuzzleNationers, this is the Sixes puzzle.

0 0 0 = 6

1 1 1 = 6

2 2 2 = 6

3 3 3 = 6

4 4 4 = 6

5 5 5 = 6

6 6 6 = 6

7 7 7 = 6

8 8 8 = 6

9 9 9 = 6

The goal is to make all of the above equations true by adding mathematical symbols.

And, naturally, there are rules.

  • You can use as many mathematical symbols as you want for each equation
  • You are not allowed to use letters, so spelling out functions like “cos” is out
  • You are not allowed to add digits of any kind, so no turning a “2” into a “12”
  • The result has to be exactly 6 ( not 6.0000000000000001 or 5.999999999999999 )
  • Square roots are allowed
  • You are not allowed to change “=” to “≠” (not equal to) or manipulate the result in any way

Take a crack at it, then scroll down past the dice to see how it’s done.

Last warning before answers!

[Image courtesy of The Progzilla Files.]

Let’s start by knocking out the easiest one.

2 2 2 = 6 can be resolved as 2+2+2=6.

6 6 6 = 6 can be resolved as 6+6-6=6 or 6-6+6=6.

They get a little trickier from here, involving multiple operations.

3 3 3 = 6 can be resolved as 3×3-3=6, becoming 9-3=6.

Fractions also come into play for a few of these equations. (This can also be represented as division.)

5 5 5 = 6 can be resolved as 5+(5/5)=6, becoming 5+1=6.

7 7 7 = 6 can be resolved as 7-(7/7)=6, becoming 7-1=6.

[Image courtesy of TAF.org.]

Okay, we’re halfway there, and now the square root rule gives us a hint regarding how to resolve the 4 equation.

4 4 4 = 6 can be 4+4-(√4)=6, becoming 8-2=6.

And this formula gives us a way to crack the 8 equation.

8 8 8 = 6 can be resolved as 8-√(√(8+8))=6, becoming 8-√(√16)=6, becoming 8-(√4)=6, becoming 8-2=6.

Square roots also come into play in solving the 9 equation.

9 9 9 = 6 can be resolved as (9+9)/(√9)=6, becoming 18/3=6.

Now, admittedly, at this point, I was stumped. I had two equations left, and no ideas regarding how to proceed.

0 0 0 = 6

1 1 1 = 6

So, I reached out to a mathematician pal of mine — the same one who helped me crack the diabolical Seesaw Puzzle from Brooklyn Nine-Nine — and he immediately knew what to do: use an exclamation point.

In mathematics, an exclamation point represents a factorial, the product of every positive number between the given number and zero.

For instance, 6! represents 6x5x4x3x2x1, or 720.

1 1 1 = 6 can now be resolved as (1+1+1)!=6, becoming 3!=6, becoming 3x2x1=6.

But what about 0 0 0 = 6?

Factorials to the rescue again! You see, 0! equals 1. So we can use the 1 equation as a template for this one.

0 0 0 = 6 can be resolved as (0!+0!+0!)!=6, becoming (1+1+1)!=6, becoming 3!=6, becoming 3x2x1=6.

And there you have it, the Sixes puzzle conquered with nothing but crafty math and puzzly skills. An excellent start to a new year of brain teaser challenges!


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The GCHQ Strikes Again!

Last year, one of the toughest puzzles I encountered all year was the GCHQ Christmas Card.

The GCHQ — Government Communications Headquarters — provides security and intelligence services for the British government. Back when they were known as GC&CS — Government Code and Cypher School — they were responsible for funding Bletchley Park and its successes cracking the German “Enigma” code during World War II.

And last Christmas, they released a puzzly Christmas card that challenged the staunchest puzzlers, with over 600,000 submissions, but only 3 successful solutions!

This year, they’re doing things a little bit differently.

[Click here for a larger version.]

This puzzle is the first step in a larger event that the GCHQ expects will take MONTHS to solve. (Their official due date for submissions is February 28, 2017!)

The big change here is that instead of a series of webpages available free to the public, they’re releasing this puzzly master challenge as a book containing over 140 codes and puzzles.

But it’s for a good cause, as “all GCHQ’s proceeds from sales of the book will be donated to the Heads Together campaign, supported by HRH The Duchess of Cambridge, who wrote the foreword for The GCHQ Puzzle Book.”

The Duchess writes:

I have always been immensely proud of my grandmother, Valerie Glassborow, who worked at Bletchley Park during the Second World War. She and her twin sister, Mary, served with thousands of other young women as part of the great Allied effort to break enemy codes. They hardly ever talked about their wartime service, but we now know just how important the men and women of Bletchley Park were, as they tackled some of the hardest problems facing the country.

In a new century, their successors at GCHQ continue this intellectual tradition. Like their Bletchley predecessors, they have become well known for valuing and understanding the importance of mental wellbeing. This is so important when dealing with such discretion and the pressure which comes with this.

William, Harry and I are very grateful that this book is supporting our Heads Together Campaign. I hope it will not only amuse and challenge readers, but help to promote an open discussion of mental health problems, which can affect anyone, regardless of age or background. Together, we are aiming to change the national conversation around mental health from stigma and fear to openness and understanding. Those who buy this book and support the Heads Together campaign will be playing a part in helping people get the important mental health care they deserve.

Puzzles and charitable works: a perfect holiday match, to be sure.

For US shoppers, the book is available as a Kindle ebook through Amazon.com, but if you want a paperback copy, you’re better off ordering it from Amazon.co.uk. With shipping and conversion, it still cost me less than $20.


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Cracking Einstein’s Riddle!

Fellow puzzlers and PuzzleNationers, it’s time to put on our thinking caps and crack another puzzly mystery!

I’ve seen this riddle making the rounds online lately, and like many logic problems, puzzles, and brain teasers that go viral, it claims that 98% of the world couldn’t solve it!

Well, I don’t buy that. (I also don’t buy that Einstein created it or had anything to do with it, but since it circulates under the name “Einstein’s Riddle,” I’m also calling it such.)

[Image courtesy of Mental Floss.]

Einstein’s Riddle

  • There are five houses that are each a different color.
  • There is a person of a different nationality in each house.
  • The five owners drink a certain drink. They each smoke a certain brand of cigar and also have a certain pet. No owner has the same pet, smokes the same brand of cigars, nor drinks the same drink.

CLUES

1. The British man lives in the red house.
2. The Swedish man has a dog for a pet.
3. The Danish man drinks tea.
4. The green house is on the left of the white house.
5. The owner of the green house drinks coffee.
6. The person that smokes Pall Mall raises birds.
7. The owner of the yellow house smokes Dunhill.
8. The person that lives in the center house drinks milk.
9. The Norwegian lives in the first house.
10. The person that smokes Blend lives next to the one that has a cat.
11. The person that has a horse lives next to the one that smokes Dunhill.
12. The one that smokes Bluemaster drinks beer.
13. The German smokes Prince.
14. The Norwegian lives next to a blue house.
15. The man that smokes Blend has a neighbor that drinks water.

The question is “who owns the fish?”


Now, the first step is going through the clues and getting all five options for every variable. This will help us with the second step: building a grid to help us organize information.

  • Colors: Blue, Green, Red, White, Yellow
  • Nationalities: British, Danish, German, Norwegian, Swedish
  • Beverages: Beer, Coffee, Milk, Tea, Water
  • Cigars: Blend, Bluemaster, Dunhill, Pall Mall, Prince
  • Pets: Birds, Cat, Dog, Fish Horse

Okay, let’s build our grid. Now, we could list every intersection of information, like a full logic problem grid, but I don’t think that’s necessary here. We can simplify.

So where do we start? Well, since several clues refer to a first house, a center house, or neighboring houses, let’s assume that we’re talking about five houses in a row, and use that as our top line. Then we can list all of the other categories we need to determine along the left side.

Now let’s fill in what we know from the clues. We know the Norwegian lives in the first house (clue #9), the man living in the center house drinks milk (clue #8), and the Norwegian lives next to the blue house (clue #14).

This might not seem like much, but we can already determine what color the Norwegian’s house is. The British man lives in the red house (clue #1), so the Norwegian’s house isn’t red. The green house is on the left of the white house (clue #4), so the Norwegian’s house is neither green nor white, since there’s no house to the left of the Norwegian’s house, and the house to the right is blue. Therefore, the Norwegian’s house is yellow, the only color left.

And that tells us something else. The owner of the yellow house smokes Dunhill (clue #7), so we know what the Norwegian smokes. Plus, the man who keeps the horse lives next to the man who smokes Dunhill (clue #11), so we know the pet for the blue house.

We can now determine the color for every house. The green house is to the left of the white house (clue #4), and the owner of the green house drinks coffee (clue #5). But since we already know the owner of the center house drinks milk, that means the green house has to be the fourth, and the white house the fifth. Therefore, the center house is red.

But that’s not all we know now! The British man lives in the red house (clue #1), so we can place him as well.

Hmmm, where do we go from here? Well, let’s take a look at the beverages. We know the Norwegian doesn’t drink coffee or milk, but we also know that the Danish man drinks tea (clue #3), so the Norwegian doesn’t. We also know that the one that smokes Bluemaster drinks beer (clue #12), and the Norwegian smokes Dunhill, so that eliminates beer. Therefore, the Norwegian drinks water.

Since the man that smokes Blend has a neighbor that drinks water (clue #15), we can place Blend in the second house. (Clue #10 tells us that the person that smokes Blend lives next to the one that has a cat, but right now, we can’t be sure if that’s the first house or the center house, so let’s table that clue for now.)

But remember clue #12, the one that smokes Bluemaster drinks beer? Well, we know all the drinks except the second house and the fifth house, and the owner of second house smokes Blend, so the owner of the fifth house must both smoke Bluemaster and drink beer.

That leaves tea as the only possible beverage for the second house, and the Danish man drinks tea (clue #3), so we can place him as well.

Now we’re cooking! The German smokes Prince (clue #13), and since we know the nationality of the third house’s owner and what the fifth house’s owner smokes, the only option remaining is the fourth house. By process of elimination, that also places the Swedish man in the fifth house and Pall Mall in the center house.

And we’re left with only the pets to place. Appropriate, given that the question that got us started is “who owns the fish?”

According to clue #2, the Swedish man has a dog for a pet, and the person that smokes Pall Mall raises birds, so that takes care of the center and fifth houses, leaving only the first house and the fourth.

Clue #10, the one we put aside earlier, now comes into play. The person that smokes Blend lives next to the one that has a cat, and since the neighbor on one side (the center house) raises birds, that leaves only the first house open to have a cat.

Therefore, the German man — who smokes Prince, drinks coffee, and lives in the green house — also owns a fish.

98% of people can’t solve it? Apparently, I have greater faith in solvers than “Einstein” did.


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It’s Follow-Up Friday: Math Puzzle Madness edition!

Welcome to Follow-Up Friday!

By this time, you know the drill. Follow-Up Friday is a chance for us to revisit the subjects of previous posts and bring the PuzzleNation audience up to speed on all things puzzly.

And today, I’d like to revisit the world of viral puzzles and discuss two that have been making the rounds on Facebook recently.

If you’ve been on social media recently, you’ve no doubt seen one or both of these puzzles:

math

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Each was probably accompanied by some hyperbolic phrasing like “95% of people and most dogs can’t solve this puzzle! Heck, they can’t even agree on an answer! CAN YOU?!?!?!?!”

Well, duh. Of course they can’t agree on an answer. There’s plenty of room to make different assumptions.

Let’s look at the first puzzle again.

math

Now, if you take the puzzle at face value, the chain would appear to be this:

1 + 4 = 5

2 + 5 (+5) = 12 (We’ve added the previous answer, which is where the +5 comes from.

3 + 6 (+12) = 21

8 + 11 (+21) = 40

So the answer is 40.

But wait. if you assume that the pattern continues for the digits between 3 and 8, you end up with this:

1 + 4 = 5

2 + 5 (+5) = 12

3 + 6 (+12) = 21

4 + 7 (+21) = 32

5 + 8 (+32) = 45

6 + 9 (+45) = 60

7 + 10 (+60) = 77

8 + 11 (+77) = 96

And, in truth, it could be either. You’re not given enough information to know for sure how to proceed. It’s a coin toss whether the last line immediately follows the third line, or whether there’s a whole bunch of lines in between and you need to “get the pattern” to extrapolate the 8th line.

Now let’s look at that second puzzle again.

14962617_1223950097651735_5609824945112529117_n

This one also has the potential for alternate answers, but instead of inferences, it depends on whether you follow the traditional order of operations (parentheses, exponents, multiplication/division, addition/subtraction) or you simply read left to right.

If you use traditional order of operations, you end up with:

Horse + Horse + Horse = 30, so Horse = 10.

Horse + Horseshoes + Horseshoes = 18, so Horseshoes = 4 and Horseshoe = 2.

Horseshoes – Boots = 2, so Boots = 2 and Boot = 1.

Boot + Horse x Horseshoe = Boot + (Horse x Horseshoe) = 1 + (10 x 2) = 21.

But if you simply read the last equation from left to right, you end up with:

Boot + Horse x Horseshoe = 1 + 10 x 2 = 11 x 2 = 22.

So, in fairness, there is no right answer to either puzzle, given the information we have.

Which, to me, doesn’t seem like a great puzzle, but it probably makes for great clickbait.


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Halloween Hallowhimsy!

Hello there, fellow puzzlers and PuzzleNationers!

It’s Halloween, that spookiest of holidays, and I’ve got two puzzly videos loaded with today’s traditional tricks and treats!

The first video comes from musician and comedian Ali Spagnola, who has created a masterful mashup of 20 tunes loaded with Halloween spirit! Can you name all 20 songs?

I’m a little disappointed that Warren Zevon’s “Werewolves of London” didn’t make the cut, but otherwise, this is awesome.

The second video comes from domino artist and friend of the Blog Hevesh5, who has created a new domino chain to celebrate All Hallow’s Eve in eerie, energetic fashion!

Whether it’s ghosts or goblins, ditties or dominoes, we hope you have a marvelous Halloween!


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It’s Follow-Up Friday: This Illusion’s Got Legs edition!

Welcome to Follow-Up Friday!

By this time, you know the drill. Follow-Up Friday is a chance for us to revisit the subjects of previous posts and bring the PuzzleNation audience up to speed on all things puzzly.

And today, I’d like to return to the subject of viral optical illusions.

Last year, we had The Dress. Then, in March of this year, we had The Jacket. And in May, we asked the question How Many Girls?

Whether we’re spotting iPhones or looking for cats in woodpiles, we can’t seem to get enough of optical illusions.

And there’s a new one making the rounds recently:

[Image courtesy of TheChive.com.]

A woman named Bree tweeted this image of a pair of bare legs that look incredibly shiny. Are they false legs? Are they lotioned or wrapped in Saran wrap? What’s going on here?

I’ll give you a few moments to ponder the image before revealing the secret behind it. Because, as Bree said, “Once you see it, you can’t unsee it.”

Ready? Okay, here we go.

Like most optical illusions, the answer is startlingly simple.

[Image courtesy of TheChive.com.]

The illusion of shininess is actually the result of strategically placed streaks of white paint or toothpaste or something similar.

Pretty impressive once you really look at it, isn’t it?

This image has truly gone viral. As I write this, it’s been retweeted over 16,000 times, and liked over 20,000 times. Bree herself seems baffled by the attention the post has received.

Amazing what you can do with a bit of white paint.


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