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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Brain Teaser: A Curious Way to Tell Time

We love tackling brain teasers, riddles, math puzzles, and mind ticklers here at PuzzleNation Blog.

Our friends at Penny Dell Puzzles also enjoy putting their puzzly skills to work by posting various brain teasers..

And one of our mutual readers was hoping we could explain how to solve a puzzle that recently made the rounds on Penny’s social media accounts.

We’d be happy to!

Today, let’s take a look at a brain teaser all about time.

So, where do we begin?

Let’s start by breaking that sentence down to make it easier to parse: “The number of hours left today is half of the number of hours already passed.”

Well, a simpler way of saying that is “the number of hours already passed is twice the number of hours left.”

So if you have the number of hours left — let’s call that X — then the number of hours already passed is twice X, or 2X. Between X and 2X, that’s your entire day covered.

Sorting that out gives us the simple formula of X + 2X = 24, since there are 24 hours in a day.

That easily becomes 3X = 24, and simple division tells us that X = 8.

So X, the number of hours left today, is 8. Which means that twice X, or 16, is the number of hours already passed.

And if there are 8 hours left in the day (or 16 hours already passed), that means it’s 4 PM.


Most of the time, brain teasers are all about efficiently organizing the information we have.

That allows us to figure out how best to use that information to move forward and solve the puzzle. This is just as true with a relatively straightforward brain teaser like this as it is with a complicated logic puzzle with all sorts of pieces to put together.

It’s all about figuring out what we know, what we don’t, and how what we DO know can lead us to what we don’t.

That’s just part of the fun.


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May the Fourth Be With You!

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Hello fellow puzzlers and PuzzleNationers! It’s Star Wars Day, and what better way to celebrate than with a puzzly Star Wars brain teaser!

A fellow Star Wars fan found this logic puzzle online and tasked us with solving it. Can you unravel the fiendish Imperial plot? Let’s find out!


Rebel Roundup

The Empire came up with a brilliant plan in order to trap various members of the Rebel Alliance: creating a fake Rebel summit. Each Imperial agent involved would invite a Rebel to the summit while posing as one of the Rebels being invited.

It would have worked perfectly, except for the fact that Admiral Ozzel posed as the person that he had invited. OOPS. Courtesy of Ozzel’s bumbling, the Rebels were warned ahead of time and armed themselves, hoping to turn the tables on the Empire.

Thanks to Han Solo’s timely warning, Luke had hidden his lightsaber and a vibroknife with R2-D2 and C-3PO respectively. These extra weapons allowed the seven Rebel agents of them to escape. It also helped that Admiral Ackbar arrived last in his ship, Home One.

Each Rebel arrived in a different spaceship, but two Rebels hitched a ride with fellow agents, so only five spaceships were involved.

Answer these questions:

  • Who traveled with Leia?
  • Who traveled with Luke?
  • What vehicle did each Rebel arrive in?
  • Which Imperial invited which Rebel?
  • Who did each Imperial pose as?
  • What weapon did each Rebel carry?

Here are your clues:

1. Leia, having been warned by Han, carried a concealable Holdout Blaster. She did not arrive in an X-Wing, nor did she fly the Millennium Falcon.

2. Han wouldn’t let anyone fly his baby. Han carried his Heavy Blaster Pistol, ready to shoot the Imperial who invited him while posing as him. This naturally made Han suspicious.

3. When Admiral Ackbar saw who invited him, he put his Force Pike to the Imperial’s throat. He was not invited by Darth Vader, who had posed as R2-D2.

4. C-3PO arrived on the Tantive IV, along with another passenger. This was not the ship Lando used.

5. The Lady Luck was flown by the man invited by Admiral Piett. Its pilot, who traveled alone, carried a Blaster Rifle with him. He gambled a bit, and almost crashed into Luke’s X-Wing. The Imperial who invited him posed as Admiral Ackbar.

6. Grand Moff Tarkin invited Admiral Ackbar. He did not pose as Luke Skywalker, nor did he pose as Leia.

7. General Veers invited R2-D2. Veers posed as R2-D2’s best friend. Captain Needa did not pose as Lando.

8. Leia was led to believe that Luke invited her to the summit. Emperor Palpatine invited Luke while posing as Leia. R2-D2 delivered his weapon to the Rebel so he could keep his father busy long enough for everyone to escape.

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Good luck, fellow puzzlers! Although the puzzle is a bit easier if you’re familiar with the Star Wars Universe, any solver should be able to crack this puzzle with the clues provided!

Let us know if you solved it in the comments below! And May the Fourth Be With You!


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