Here We Come A-Wassailing, With Some Reworded Carols!


[Image courtesy of The Sun Papers.]

A few years ago, I posted a holiday puzzle that had been floating around the Internet for years. It was a list of Christmas songs and carols whose titles had been reworded, and it was up to the reader to identify the actual titles.

It was a popular post, but something about the list always bothered me. There were 21 reworded titles, which didn’t strike me as very Christmassy at all. I mean, why not 12? Or 24? Or, heck, 25?

So, I did something about it. I added 10 new reworded titles to the list, bringing the total to 31, one for every day in December. Let’s see how many PuzzleNationers can crack all 31 titles, shall we? Enjoy!

1.) Move hitherward the entire assembly of those who are loyal in their belief.

2.) Listen, the celestial messengers produce harmonious sounds.

3.) Proceed forth declaring upon a specific geological alpine formation.

4.) Nocturnal timespan of unbroken quietness.

5.) Embellish the interior passageways.

6.) An emotion excited by the acquisition or expectation of good given to the terrestial sphere.

7.) Twelve o’clock on a clement night witnessed its arrival.

8.) The Christmas preceding all others.

9.) Small municipality in Judea southeast of Jerusalem.

10.) In a distant location the existence of an improvised unit of newborn children’s slumber furnishings.

11.) Tintinnabulation of vacillating pendulums in inverted, metallic, resonant cups.

12.) The first person nominative plural of a triumvirate of far eastern heads of state.

13.) Geographic state of fantasy during the season of Mother Nature’s dormancy.

14.) In awe of the nocturnal timespan characterized by religiosity.

15.) Natal celebration devoid of color, rather albino, as an hallucinatory phenomenon for me.

16.) Expectation of arrival to populated areas by mythical, masculine perennial gift-giver.

17.) Obese personification fabricated of compressed mounds of frozen minute crystals.

18.) Tranquility upon the terrestial sphere.

19.) Omnipotent supreme being who elicits respite to ecstatic distinguished males.

20.) Diminutive masculine master of skin-covered percussionistic cylinders.

21.) Jovial Yuletide desired for the second person singular or plural by us.

22.) Allow winter precipitation in the form of atmospheric water vapor in crystalline form to descend.

23.) A first-person observer witnessed a female progenitor engaging in osculation with a hirsute nocturnal intruder.

24.) Your continued presence remains the sole Yuletide request of the speaker in question.

25.) Permanent domicile during multiple specific celebratory periods.

26.) Diminutive person regarded as holy or virtuous known by the informal moniker shared by two former Russian tsars.

27.) More than a passing resemblance to an annual winter festival is emerging.

28.) Are you registering the same auditory phenomenon I am currently experiencing?

29.) Overhead at the summit of the suburban residence.

30.) Attractive or otherwise visually pleasing wood pulp product.

31.) Parasitic European shrub accompanied by a plant with prickly green leaves and baccate qualities.

How many did you unravel, fellow puzzlers? Let us know in the comments section below! We’d love to hear from you!

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Riddles, Riddles Everywhere!

I’ve had riddles on the brain recently, because I keep seeing them everywhere. Over the last few weeks, they’ve popped up in games, TV shows, books, and even emails to the blog.

It all started with our twice-monthly office D&D game. Every other Thursday, a group of us commandeers one of the conference rooms at lunchtime and enjoys an hour of dice-fueled storytelling, adventure, and fun.

As is often the case with a fantasy-inspired game, there was a river to cross and a riddle to answer in order to pass.

A murderer is condemned to death. He has to choose between three rooms. The first is full of raging fires, the second is full of assassins with loaded guns, and the third is full of lions that haven’t eaten in a year. Which room is safest for him?

This is a classic riddle, usually titled “Three Doors” or “The Murderer’s Riddle.”


And when you’ve got a team of puzzle solvers in your D&D group, this riddle is no challenge at all.

(If you’re curious about the solution, you pick door #3. After a year of not eating, the lions would be dead, so it would be safe to enter that room.)

Later on in the game, we again had to barter passage across a body of water, either answering a riddle or battling a demon to the death.

Naturally, we chose the riddle.

What is the creature that walks on four legs in the morning, two legs at noon and three in the evening?

This is another classic riddle — the Riddle of the Sphinx, most famously solved by Oedipus — and posed no challenge to our merry band of misfit adventurers.

(If you don’t know this one, the answer is “man,” since you walk on four legs as a child, aka crawling, two legs as an adult, and with a cane when you’re older. The day — morning, noon, and evening — represents a lifetime.)

We crossed the lake, and our adventure continued, and I thought I was done with riddles for a bit.

Then a few days later, I got caught up on the latest season of MTV’s The Challenge, a reality/competition game show. (I’ve written about some of their puzzly challenges in the past.)

And, wouldn’t you know it, this week’s challenge involved a riddle.


Both teams would start on this platform, sending pairs of swimmers out on a long swim to retrieve keys. Those four keys would then open both a chest full of letter tiles and a riddle to be solved. The first team to solve the riddle with the letters available would win the challenge.

Once all the drama of selecting partners — given that many of the players weren’t strong swimmers, and the slowest-swimming team would be eliminated from the game — there was plenty of tension to be had.

But finally, all four keys were retrieved by the teams, and the riddle revealed:


I am a 5 letter word.

I am normally below you.

If you remove my 1st letter, you’ll find me above you.

If you remove my 1st and 2nd letters, you can’t see me.

The teams were initially baffled, playing around with different words and various combinations of letter tiles in the hopes that it would spark something.


Eventually, competitor Ashley came up with a three-letter word that you couldn’t see — AIR — and her team quickly came up with the correct answer: CHAIR.

(A chair is normally below you, hair is above you (sorta), and air can’t be seen.)

So, three riddles in a matter of days. It’s officially a pattern. And so far, I’m three for three on solving these riddles.

A week or so later, though, yet another riddle arrived, this time by email. And I admit, I’m a little stumped.

What has a bell but isn’t a church. Is full of air but is not a balloon?

What do you think, fellow puzzlers and PuzzleNationers? Any ideas? Let me know in the comments section below. I have a few theories, but nothing that feels like a conclusive answer.

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The Beale Ciphers: A Puzzly Treasure Hidden Since the 1800’s?


There’s nothing quite like a treasure hunt to spark the imagination. From The Treasure of the Sierra Madre to the adventures of Indiana Jones, from tales as far back as Poe’s “The Gold-Bug” to stories as recent as an episode of NCIS: New Orleans last year, a treasure hunt can turn a crime story or an adventure tale into an irresistible narrative for the ages.

Thankfully, there are a few treasure hunts lurking out there in the real world, offering clever solvers the chance to live out their own adventure. In the past, we’ve explored the mystery of Forrest Fenn’s Rocky Mountain treasure, we’ve chronicled efforts to locate all of Byron Preiss’s The Secret treasures, and we’ve suggested tactics for cracking Jason Rohrer’s A Game for Someone hunt.

But as intriguing as those hunts are, none of them have spanned more than a century of searching. (Without resulting in unfortunate demises, that is. We’re looking at you, Oak Island.)

No, that singular honor belongs to a treasure hunt known as the Beale Ciphers.


As the story goes, a man named Thomas J. Beale buried a treasure trove of gold and silver somewhere in Bedford County, Virginia, in the early 1800s. Beale then encrypted the location of the treasure, the contents of the treasure, and the names of those he wished to have the treasure. Beale handed off those encryptions to an innkeeper, then vanished, never to be seen again. (His promise of later providing the key for the ciphers was never fulfilled.)

The innkeeper failed to crack the ciphers, then held onto them for decades before passing them along to an unnamed friend before his death. The unnamed friend spent twenty more years trying to unravel the encryptions (managing to solve the second of the three encrypted messages). Eventually, the friend published the encryptions and the story of Beale’s treasure in a pamphlet he began selling in 1885.

So, how do the ciphers work?

It’s simple, really. Take a book, pick a given page, and number all of the words on the page. (Or just start at the beginning of the book.)

If you’re using A Tale of Two Cities, for instance:

1 It
2 was
3 the
4 best
5 of
6 times,
7 it
8 was
9 the
10 worst
11 of
12 times…

So, using the first letters of each word (and the corresponding number), the word BOW could be encrypted 4 11 8 or 4 11 2 or 4 11 10.

This grants people in the know two advantages. The code is incredibly difficult to break on its own, because unlike a cryptogram (or any other message encrypted with a Caesar cipher or a one-to-one relationship between coded letters), each appearance of a given letter could be a different number, not the same one over and over.

Plus, if you know the key (the book and page number), decoding it requires no puzzly skill at all.

It’s diabolical and effective, as proven by Beale’s trio of ciphers, since only one has been cracked (because the solver stumbled upon the Declaration of Independence as the key).

[The second Beale cipher.]

The decrypted text from the second cipher:

I have deposited in the county of Bedford, about four miles from Buford’s, in an excavation or vault, six feet below the surface of the ground, the following articles, belonging jointly to the parties whose names are given in number three, herewith:

The first deposit consisted of ten hundred and fourteen pounds of gold, and thirty-eight hundred and twelve pounds of silver, deposited Nov. eighteen nineteen. The second was made Dec. eighteen twenty-one, and consisted of nineteen hundred and seven pounds of gold, and twelve hundred and eighty-eight of silver; also jewels, obtained in St. Louis in exchange to save transportation, and valued at thirteen thousand dollars.

The above is securely packed in iron pots, with iron covers. The vault is roughly lined with stone, and the vessels rest on solid stone, and are covered with others. Paper number one describes the exact locality of the vault, so that no difficulty will be had in finding it.


Of course, there are some problems there, even with the cipher that treasure hunters consider solved. You see, there are some irregularities with the solution. Not only are there four misspellings in the translation, but a variation on the original Declaration of Independence must be used or the cipher doesn’t decode correctly.

Now, mistakes happen. (As we learned with the story of Brian Patrick Regan.) But if there are mistakes in the two unsolved ciphers as well, that only makes the chances of finding the proper key even slimmer, because a mistake in the early numbers of the code might convince someone that they’ve got the wrong key, even if they have the right one!

Do you find that challenge daunting, fellow puzzlers? It’s understandable if you do. The other two ciphers have resisted the best efforts of even master cryptographers and cryptanalysts.

Given that the Declaration of Independence was the key for the second cipher, many aspiring treasure hunters have tried using other famous historical documents as possible keys for the other ciphers, including the Magna Carta, the Constitution, the Monroe Doctrine, and more, as well as the plays of Shakespeare and the Lord’s Prayer.


There are also plenty of reasons to doubt that this treasure exists at all. (The same question marks hang over some of the other treasure hunts we’ve mentioned, like Forrest Fenn’s.)

There are questions regarding the language in the pamphlet, where the gold was supposedly found, why Beale would bother encrypting the names of the people he wanted to inherit the treasure, and even whether Beale himself ever existed in the first place. (Famous skeptic and investigator of the supernatural Joe Nickell believes the pamphlet is a fraud.)

But does that mean the ciphers are? Not necessarily.

An analysis in 1970 by Dr. Carl Hammer of Sperry-UNIVAC indicated that the number patterns are not random. He believed that further attempts at cracking the ciphers would be worthwhile.

Heck, even our old codebreaking friends Elizebeth Smith Friedman and her husband William tried to unravel the Beale ciphers, but without success. She called the ciphers “a diabolical ingenuity designed to lure the unwary reader.”

And, of course, not every hunter has come away empty-handed. One team of treasure hunters stumbled upon a cache of Civil War artifacts while hunting for Beale’s trove.

So what do you think, PuzzleNationers? Is the Beale treasure real? Will it ever be found? Let us know in the comments section below! We’d love to hear from you.

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A Clued Sudoku Puzzle? It’s More Than Meets the Eye


The hunt is always on for the next big puzzle idea.

Sometimes, it’s an old idea that gets repackaged and catches fire. That’s what happened with Sudoku, a puzzle that had been around since the late ’70s, but only rose to prominence decades later.

Other times, it’s a combination of different puzzle types that yields something special. Our friends at Penny Dell Puzzles, for instance, have a popular puzzle called Anagram Magic Square, which combines crossword-style cluing, anagrams, and the mathematical element of a magic square to create an engaging puzzle experience you can solve from several angles.

Whether a puzzle is destined for superstardom or not depends on a lot of factors: difficulty, the type of solving it involves, how intuitive the solving is (i.e. needing a lengthy explanation vs. getting the gist of the puzzle from a glance), visual aesthetics, and more.

As a puzzler, it’s always exciting to try out a new puzzle. Wholly original ideas are rare, to be sure, but even a single twist on an old classic can be enjoyable if executed well.

Today, we’re taking a look at a puzzle that combines Sudoku with cryptic crosswords (aka British-style crosswords). It’s called Cluedoku, and it was created by cryptic constructor Charlie Methven, better known in solving circles as Chameleon, a contributor to British puzzle outlets like The Guardian.


[Just a sample of the puzzle. Check out the entire puzzle here.]

Like Sudoku, Cluedoku involves placing the digits 1 through 9 into each row, column, and 3×3 square in the grid. But unlike Sudoku, there are no set letters.

Instead, you have 81 clues, one for every cell in the grid, utilizing cryptic-style cluing to hint toward which of the nine numbers goes in a given cell.

Once you’ve unraveled a clue and placed a number in the grid, standard Sudoku rules apply: that number will only appear once in a row, column, or 3×3 square.

But that’s easier said than done. These clues run the gamut of slyly clever to almost baffling. Even when you consider that there are only nine possible answers for each clue, it’s still a challenge. (Plus, not all of the clues adhere to the standard cryptic cluing mechanic of having both a definition AND a wordplay clue included.)

That being said, you’ll find lots of traditional cryptic cluing tricks at play here.

Now, we’re going to be discussing specific clues and answers from this puzzle, so this is your spoiler warning.








Last chance to solve without spoilers!







Okay, here we go.

In terms of traditional cryptic cluing gimmickry, we see hidden words, anagrams, homophones, word reversals, and more.

In clue 6 — Axis revolves without beginning to accelerate — we revolve (aka reverse) axis to spell SIXA, and then drop the A (“without beginning to accelerate”) to spell SIX.

In clue 8 — Prime cut from sloth reeks — the answer hides in plain sight, as a prime number (three) reads out in sloTH REEks (and can be cut out of it).

In clue 22 — Scenes in X-Men Origins reveal how many claws Wolverine has! — the phrase “origins reveal” points towards the first letters of the words that precede it proving the answer, meaning that SIX is the number of claws Wolverine has (three on each hand).

There is a similar game in clue 67 — With only seconds remaining, Officer Columbo outwits crook — which has the second digits of “Officer Columbo outwits crook” spelling out FOUR.

In clue 27 — UFO demolished third of Parliament Square — the letter R (“third of Parliament”) gets mixed up with UFO to make FOUR, a square.


But other clues would be familiar to crossword solvers in America.

Clue 29 — Number of Romans in the New Testament? — is simple wordplay for 6, since Romans is the SIXth book. (Similarly, clue 62 — Number of lines taken by bar staff — is a reference to the FIVE lines that make up a staff in sheet music.)

Clue 34 — Top score in Scrabble — is a bit more devious, requiring you to know that T is worth 1 point, O is worth 1 point, and P is worth 3 points, making the correct answer FIVE.

Clue 48 — Man’s arms’ legs’ digit — feels like a clue you’d see at the Indie 500 or Lollapuzzoola, because it’s initially baffling, but then reveals itself as merely clever and challenging. You see, there are THREE legs on the coat of arms for the Isle of Man. But that’s concealed by the wordplay involving three different words that don’t mean what you’d think.

This mix of American and British-style clues made for a fun solve that mixed and mingled two worlds of cluing nicely.

I think my favorite clue was Clue 39 — 192+284 — because it was built like one of those magazine word puzzles, the ones where “rockcaughthardplace” means “caught between a rock and a hard place.” In this case, you have “2+2” literally in 1984. And for anyone familiar with George Orwell’s famous novel, 2+2 in 1984 equalled FIVE.

Although obviously Cluedoku isn’t really sustainable as a recurring puzzle — you’d burn out your anagrams and homophones pretty quickly, as Chameleon himself stated in an interview — it is an impressive marriage of two different puzzles that rarely interact otherwise.

But he did raise the possibility of another variation in the future:

If I did another Chameleon cluedoku, I think I’d use the seven colours of the rainbow plus black and white, as solvers could then colour in each square as they solved. How’s “Cry over Norwich’s core Canary”?

That sounds like a fun follow-up to an interesting puzzle.

What did you think of Cluedoku, fellow puzzlers and PuzzleNationers? Let us know in the comments below! We’d love to hear from you.

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Putting Clever Cluing to the Test?


As a puzzler, there are few article titles that serve as more efficient clickbait than “6 CHALLENGING CROSSWORD PUZZLE CLUES THAT WILL LEAVE YOU CLUELESS,” so when I saw that title, fellow puzzlers, you know I clicked.

This article by the crew at Wealth Words claims to offer “the trickiest clues that have ever existed.” That is quite a bold statement. Shall we try our luck and see how we do?

Now, before we start, it’s worth noting that we’re at a huge disadvantage here, because any clue, easy or tough, can be made easier if you know some of the letters in the word thanks to words in the grid you’ve already placed that cross this particular entry. We don’t have any of those helper letters, so we’re going to have to rely solely on our sharp wits, wordplay skills, and love of punnery.

Okay, let’s get to it.

Clue #1: Leaning column? (9 letters)

Most crossword fans know that a question mark virtually always means there’s wordplay afoot, so you know you can’t take this clue at face value, which means anything relating to Pisa is probably out. If you focus on “leaning,” that could take you anywhere from Jenga to drunkenness, so let’s play with “column.” Other columns appear in graphs, Excel files, and newspapers.

A-ha. Newspaper column. And some of those “lean” to either the left or the right, depending on the author. This train of thought leads us to the intended answer OPED PIECE.

Grade: A- (It’s a solid clue where the answer doesn’t necessarily immediately jump out at you, but makes total sense once you’ve puzzled it out.)

Clue #2: Strips in a club (5 letters)

[Now, to be fair, “club” was capitalized in the clue on the webpage, but I felt like that was misleading, so I fixed it here. After all, capitalization can be used to great effect in crafty cluing — particularly if you conceal the capital word by making it the first word in the clue, which is always capitalized regardless — but here, it becomes an unnecessary red herring.]

This one is slightly harder, because you don’t immediately get the hint that there’s wordplay involved, since there’s no question mark.

This is one of my favorite kinds of clever cluing, the sort where our preconceived notions of word forms works against us. (Also, it sounds naughty, but isn’t, which I also quite enjoy.) At first glance here, the phrasing makes it sound like “strips” is a verb, when it’s really a plural noun.

And once you get into that mindset, you realize that we’re not talking about that kind of club, and the intended answer emerges: BACON.

Grade: A (Misdirection plus a tongue-in-cheek bit of lewdness? Great stuff.)

Clue #3: Group of crows (6 letters)

I have no idea how this one made it onto the list. Anyone who knows their animal groupings knows that a group of crows is a MURDER. There’s no tricky cluing or misdirection here, just something that might not be in the common knowledge. (But again, I think people are more likely to come up with this one that “exaltation of larks,” “smack of jellyfish,” or “parliament of owls.”)

Grade: D- (Could be difficult for some solvers, but only for unfamiliarity, not style.)

Clue #4: “Yep, perfectly clear” (7 letters)

Okay, this one has quotations around it, which both means it’s a spoken line and it’s likely non-standard, so you won’t find it in a dictionary. It’s probably a phrase, and used in casual conversation.

The answer, as it turns out, is I HEAR YA, which I don’t think any solver would come up with unless they had a few crossing letters filled in for them. The slangy spelling of YA and the informal wording altogether pretty much precludes this from being a “see-it-and-get-it” sorta clue.

Grade: C

Clue #5: [Boo-Hoo] (5 letters)

Brackets are used less commonly than quotation marks or question marks in crossword clue, so it’s more likely that a casual solver wouldn’t immediately recognize what to do with this clue. Usually, brackets indicate this is a non-traditional clue, either making an oblique reference to something or indicating it’s a non-verbal clue like a cough.

In this case, this is meant to be the actual sound of someone crying or something of that nature. So it could be something informal like CRYIN’ or TEARS (as opposed to the more traditional “in tears”) or something like that.

As it turns out, they were looking for I’M SAD. Which is pretty blah. It’s not a standard phrase, and comes off as a cheaply constructed way out of a bad corner, not a solid bit of fill to keep the puzzle interesting.

Grade: F

Clue #6: They come in last (3 letters)

This clue is fairly tough, because it’s both vaguely worded and has a curious letter count. It’s plural phrasing (with “they”), so that immediately makes you want to tack an S onto the end of the word. But it’s also such a short entry that a two-letter word plus S doesn’t seem to fit the clue.

So what comes in last? “End” would fit, if not for the plural phrasing. “P.S.” comes in last, but “P.S.’S” is really clunky, and I don’t recall ever seeing that pluralized.

So what were they looking for? XYZ. Ah. Alphabet entries. You’ll usually see entries like this centering around the first three letters (ABC) or a random string (“RST” seems to come up more often than most), and XYZ certainly fits the bill. But, in the end, it’s not a real entry, and it feels a little cheap, despite the decent wordplay involved in the cluing.

Grade: C-

So, what did I think of the Wealth Words “6 CHALLENGING CROSSWORD PUZZLE CLUES THAT WILL LEAVE YOU CLUELESS” challenge?

I thought it started off very strong with two clever, slippery clues that required you to play with the words and come at them from several angles before stumbling upon the correct solution, and I quite enjoyed those clues.

But the quiz took a real nose dive in quality starting with Clue #3, which had no wordplay at all. #4 and #5 relied heavily on being slangy non-standard verbiage rather than adept cluing or creative fill, and #6 was a bit of a cop-out, even if the cluing quality rebounded nicely.

All in all, I thought the specious entries outweighed the clever cluing on display early on, making for an underwhelming set of clues.

Final grade: C+.

What did you think of these group of challenging clues, fellow puzzlers and PuzzleNationers? Did you enjoy them or find them wanting? Let us know in the comment section below, we’d love to hear from you!

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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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