A Coin Puzzle: My Two Cents (Plus 97 More)

Our friends at Penny Dell Puzzles recently shared the following brain teaser on their social media:

Naturally, we accepted the challenge.

Now, before we get started with this one, we have to add one detail: which coins we’re allowed to use. It’s safe to assume that pennies, nickels, dimes, and quarters are available, but the question doesn’t say anything about half-dollar coins.

So we’re going to figure out the correct answer without half-dollar coins available, and then with half-dollar coins available.

Let’s begin.


[Image courtesy of How Stuff Works.]

The easiest way to get started is to figure out the smallest number of coins we need to make 99 cents, since that’s the highest number we need to be able to form. Once we have that info, we can work backwards and make sure all the other numbers are covered.

For 99 cents, you need 3 quarters, 2 dimes, and 4 pennies. That’s 25 + 25 + 25 + 10 + 10 + 1 + 1 + 1 + 1 = 99.

Right away, we know we’re close with these 9 coins.

You don’t need more than 3 quarters, for instance, because your possible totals are all below $1.

Now, let’s make sure we can form the numbers 1 through 24 with our chosen coins. (If we can, we’re done, because once we’ve covered 1 through 24, we can simply add one quarter or two quarters to cover 25 through 99.)

Our four pennies cover us for 1 through 4. But wait, there’s 5. And we can’t make 5 cents change with 4 pennies or 2 dimes. In fact, we can’t make 5, 6, 7, 8, or 9 cents change without a nickel.

So let’s add a nickel to our current coin count. That makes 3 quarters, 2 dimes, 1 nickel, and 4 pennies. (Why just 1 nickel? Well, we don’t need two, because that’s covered by a single dime.)

Our four pennies cover 1 through 4. Our nickel and four pennies cover 5 through 9. Our dime, nickel, and four pennies cover 1 through 19. And our two dimes, one nickel, and four pennies cover 1 through 29. (But, again, we only need them to cover 1 through 24, because at that point, our quarters become useful.)

That’s all 99 possibilities — 1 through 99 — covered by just ten coins.

[Image courtesy of Wikipedia.]

But what about that half-dollar?

Well, we can apply the same thinking to a coin count with a half-dollar. For 99 cents, you need 1 half-dollar, 1 quarter, 2 dimes, and 4 pennies. That’s 50 + 25 + 10 + 10 + 1 + 1 + 1 + 1 = 99.

Now, we make sure we can form the numbers 1 through 49 with our chosen coins. (Once we can, we can simply add the half-dollar to cover 50 through 99.)

Once again, we quickly discover we need that single nickel to fill in the gaps.

Our four pennies cover 1 through 4. Our nickel and four pennies cover 5 through 9. Our dime, nickel, and four pennies cover 1 through 19. Our two dimes, one nickel, and four pennies cover 1 through 29. And our one quarter, two dimes, one nickel, and four pennies cover 1 through 54. (But, again, we only need them to cover 1 through 49, because at that point, our half-dollar becomes useful.)

That’s all 99 possibilities — 1 through 99 — covered by just nine coins.


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Brain Teaser Week: Answers Edition!

Did you enjoy Brain Teaser Week, fellow puzzlers and PuzzleNationers? We certainly hope so! It was a fun experiment in dedicating an entire week to a particular type of puzzle.

We gave you three puzzles to challenge your deductive, mathematical, and puzzly skills, and now it’s time to break them down and explain them.


Tuesday’s Puzzle:

A set of football games is to be organized in a “round-robin” fashion, i.e., every participating team plays a match against every other team once and only once.

If 105 matches in total are played, how many teams participated?

If every team plays every other team once, you can easily begin charting the matches and keeping count. With 2 teams (Team A and Team B), there’s 1 match: AB. With 3 teams (A, B, and C), there are 3 matches: AB, AC, BC. With 4 teams (A, B, C, and D), there are 6 matches: AB, AC, AD, BC, BD, CD. With 5 teams (A, B, C, D, and E), there are 10 matches: AB, AC, AD, AE, BC, BD, BE, CD, CE, DE.

Now, we could continue onward, writing out all the matches until we reach 105, but if you notice, a pattern is forming. With every team added, the number of potential matches increases by one.

With one team, 0 matches. With two teams, 1 match. With three teams, 2 more matches (making 3). With four teams, 3 more matches (making 6). With five teams, 4 more matches (making 10).

So, following that pattern, 6 teams gives us 15, 7 teams gives us 21, and so on. A little simple addition tells us that 15 teams equals 105 matches.


Thursday’s Puzzle:

You want to send a valuable object to a friend securely. You have a box which can be fitted with multiple locks, and you have several locks and their corresponding keys. However, your friend does not have any keys to your locks, and if you send a key in an unlocked box, the key could be copied en route.

How can you and your friend send the object securely?

(Here’s the simplest answer we could come up with. You may very well have come up with alternatives.)

The trick is to remember that you’re not the only one who can put locks on this box.

Put the valuable object into the box, secure it with one of your locks, and send the box to your friend.

Next, have your friend attach one of his own locks and return it. When you receive it again, remove your lock and send it back. Now your friend can unlock his own lock and retrieve the object.

Voila!


Friday’s Puzzle:

The owner of a winery recently passed away. In his will, he left 21 barrels to his three sons. Seven of them are filled with wine, seven are half full, and seven are empty.

However, the wine and barrels must be split so that each son has the same number of full barrels, the same number of half-full barrels, and the same number of empty barrels.

Note that there are no measuring devices handy. How can the barrels and wine be evenly divided?

For starters, you know your end goal here: You need each set of barrels to be evenly divisible by 3 for everything to work out. And you have 21 barrels, which is divisible by 3. So you just need to move the wine around so make a pattern where each grouping (full, half-full, and empty) is also divisible by 3.

Here’s what you start with:

  • 7 full barrels
  • 7 half-full barrels
  • 7 empty barrels

Pour one of the half-full barrels into another half-full barrel. That gives you:

  • 8 full barrels
  • 5 half-full barrels
  • 8 empty barrels

If you notice, the full and empty barrels increase by one as the half-full barrels decrease by two. (Naturally, the total number of barrels doesn’t change.)

So let’s do it again. Pour one of the half-full barrels into another half-full barrel. That gives you:

  • 9 full barrels
  • 3 half-full barrels
  • 9 empty barrels

And each of those numbers is divisible by 3! Now, each son gets three full barrels, one half-full barrel, and three empty barrels.


How did you do, fellow puzzlers? Did you enjoy Brain Teaser Week? If you did, let us know and we’ll try again with another puzzle genre!

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The Conclusion of Brain Teaser Week!

It’s the third and final day of our celebration of all things brain-teasing, riddling, and word-tricky, and we’ve got one last devious challenge lined up for you.

Remember! On Tuesday, Thursday, and Friday of this week, a different brain teaser or word problem will be posted, and it’s up to you to unravel them. Contact us with the correct answer — either here on the blog through the comments, or on Twitter, Facebook, or Instagram through our messages — and you’ll be entered into a pool to win a prize!

And yes, you can enter more than once! Heck, if you solve Tuesday, Thursday, AND Friday’s puzzles, that’s three chances to win!

Let’s get started, shall we?


Here’s today’s brain teaser, which mixes the math of Tuesday’s puzzle with the deductive reasoning of Thursday’s puzzle:

The owner of a winery recently passed away. In his will, he left 21 barrels to his three sons. Seven of them are filled with wine, seven are half full, and seven are empty.

However, the wine and barrels must be split so that each son has the same number of full barrels, the same number of half-full barrels, and the same number of empty barrels.

Note that there are no measuring devices handy. How can the barrels and wine be evenly divided?


Good luck, fellow puzzlers! We’ll see you Tuesday with answers for all three brain teasers!

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Brain Teaser Week Continues!

It’s Day 2 of our celebration of all things mind-tickling, and we’ve got another diabolical challenge lined up for you.

Remember! On Tuesday, Thursday, and Friday of this week, a different brain teaser or word problem will be posted, and it’s up to you to unravel them. Contact us with the correct answer — either here on the blog through the comments, or on Twitter, Facebook, or Instagram through our messages — and you’ll be entered into a pool to win a prize!

And yes, you can enter more than once! Heck, if you solve Tuesday, Thursday, AND Friday’s puzzles, that’s three chances to win!

Let’s get started, shall we?


Here’s today’s brain teaser, which is less mathematical than Tuesday’s and more logical or deductive:

You want to send a valuable object to a friend securely. You have a box which can be fitted with multiple locks, and you have several locks and their corresponding keys. However, your friend does not have any keys to your locks, and if you send a key in an unlocked box, the key could be copied en route.

How can you and your friend send the object securely?


Good luck, fellow puzzlers! We’ll see you Friday with our next brain teaser!

Thanks for visiting PuzzleNation Blog today! Be sure to sign up for our newsletter to stay up-to-date on everything PuzzleNation!

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Welcome to Brain Teaser Week!

[Image courtesy of Bogoreducare.org.]

Hello puzzlers and PuzzleNationers!

This week I thought I would try something different and focus on a theme for the week’s posts, rather than posting about different topics on our usual days.

So, please join me in some puzzly challenges as we celebrate Brain Teasers Week here at PuzzleNation Blog.

On Tuesday, Thursday, and Friday of this week, a different brain teaser or word problem will be posted, and it’s up to you to unravel them. Contact us with the correct answer — either here on the blog through the comments, or on Twitter, Facebook, or Instagram through our messages, and you’ll be entered into a pool to win a prize!

And yes, you can enter more than once! Heck, if you solve Tuesday, Thursday, AND Friday’s puzzles, that’s three chances to win!

Let’s get started, shall we?


Here’s today’s brain teaser:

A set of football games is to be organized in a “round-robin” fashion, i.e., every participating team plays a match against every other team once and only once.

If 105 matches in total are played, how many teams participated?


Good luck, fellow puzzlers! We’ll see you Thursday with our next brain teaser!

Thanks for visiting PuzzleNation Blog today! Be sure to sign up for our newsletter to stay up-to-date on everything PuzzleNation!

You can also share your pictures with us on Instagram, friend us on Facebook, check us out on TwitterPinterest, and Tumblr, and explore the always-expanding library of PuzzleNation apps and games on our website!

Puzzles in Pop Culture: Die Hard With a Vengeance

die-hard-with-a-vengeance-original

It’s summer, and when it comes to Hollywood, summer means big blockbuster action movies. One of my favorites is the Bruce Willis / Samuel L. Jackson modern classic Die Hard With a Vengeance.

For those who are unfamiliar with the film — shame on you! — it’s the third installment of the Die Hard franchise, featuring catchphrase-spouting New York City cop John McClane battling terrorists, criminals, and all sorts of unsavory characters.

In Die Hard With a Vengeance, a bomber named Simon is terrorizing the city and McClane is one of his playthings, forced to play Simon Says and accomplish increasingly difficult tasks that Simon sets before him. As McClane (and electrician Zeus Carver, who saves McClane from the first of Simon’s games) race around the city trying to prevent other bombs from going off, Simon enacts an elaborate scheme to rob the city.

Thankfully, McClane and Zeus have a knack for brain teasers and riddles, because several of Simon’s devious tasks require quick thinking and sharp puzzle skills.

diehardwithavengeance1

[One of the last movies to feature payphones as a key plot point…]

First, Simon hits them with a math problem:

As I was going to St. Ives, I met a man with seven wives. Each wife had seven sacks, every sack had seven cats, every cat had seven kittens. Kittens, cats, sacks, wives. How many were going to St. Ives?

As McClane fervently tries to do multiplication in his head, Zeus realizes this isn’t a word problem, it’s a riddle. The man was going to St. Ives when he met this man, meaning the man was coming from St. Ives. So the wives, sacks, cats, and kittens are irrelevant. Only the narrator is going to St. Ives, so the answer to the riddle is 1.

stives-1886892

[Seems like a nice place to take your many wives…]

In their second puzzly task, Simon offers the following question:

“What has four legs and always ready to travel?”

McClane doesn’t get it, but Zeus immediately identifies it as an elephant joke for kids (although he doesn’t actually deliver the punchline: an elephant, because it has four legs and a trunk).

They quickly spot a nearby fountain with an elephant statue. Awaiting them is a suitcase bomb and two empty jugs. When McClane opens the suitcase, he accidentally arms the bomb, and Simon calls to inform them that the only way to disarm the bomb is to fill one of the jugs with exactly four gallons of water and place it on the scale in the suitcase.

die-hard-vengeance-laptop

[And they say what we learn in school has no practical, real-world applications…]

The problem is the two jugs hold 3 gallons and 5 gallons, respectively. Simon has set them up with another brain teaser, but one with a dire time limit to solve.

Thankfully, there are two ways to solve this brain teaser.

Method #1

  • Fill the 3-gallon jug and pour the water into the 5-gallon jug.
  • Refill the 3-gallon jug and pour the water into the 5-gallon jug until the 5-gallon jug is full, leaving 1 gallon in the 3-gallon jug.
  • Empty the 5-gallon jug and pour the 1 gallon of water from the 3-gallon jug into the 5-gallon jug.
  • Fill the 3-gallon jug again and empty it into the 5-gallon jug, leaving exactly 4 gallons in the 5-gallon jug.

Method #2

  • Fill the 5-gallon jug and pour that water into the 3-gallon jug until the 3-gallon jug is full, leaving 2 gallons in the 5-gallon jug.
  • Empty the 3-gallon jug and pour the 2 gallons of water from 5-gallon jug into the 3-gallon jug.
  • Refill the 5-gallon jug and pour that water into the 3-gallon jug until the 3-gallon jug is full, leaving 4 gallons in the 5-gallon jug.

Either way, you’ve disarmed the bomb. Good job!

140-billion-die-hard

[While Simon has McClane and Zeus run
all over the city, he has one specific goal…]

The final riddle Simon gives Zeus and McClane is another brain teaser masquerading as a math problem:

“What is 21 out of 42?”

At the time of the film’s release, there had been 42 presidents, so 21 out of 42 was President Chester A. Arthur, and Chester A. Arthur Elementary School was where Simon had hidden one of his bombs (a fake one, as it turns out) as a distraction.

In the end, McClane and Zeus outwit the cunning Simon, and once again, puzzle-solving skills save the day! Hooray!

Thanks for visiting PuzzleNation Blog today! You can share your pictures with us on Instagram, friend us on Facebook, check us out on TwitterPinterest, and Tumblr, and be sure to check out our library of PuzzleNation apps and games!