Unraveling the Riddle of Math Puzzles!

Math puzzles are among the most intimidating in the world of puzzles. Many people will happily dive into a crossword or tackle a word seek at a moment’s notice, but drop some numbers into a puzzle, and they hesitate.

But there’s no reason to fear!

Math puzzles are certainly a different form of puzzling, but like all puzzles, there’s always a way in, if you know how to look for it. Today, we’re going to solve two math puzzles together in the hopes of demystifying this style of puzzle.

Let’s take a look at our first math puzzle, “Count the Votes.”

A problem developed at a recent election where 5,219 votes were cast for four candidates. The victor exceeded his opponents by 22, 30, and 73 votes, yet not one of them knew how to figure out the exact number of votes received by each. Can you?

Okay, where do we begin?

Let’s start with what we know. We know the total number of votes, 5,219. That will be one side of our equation.

We also know that the winner beat his three opponents by 22 votes, 30 votes, and 73 votes, respectively. Which means that the number of votes the winner received is the key to solving this puzzle. Let’s call that number of votes “x.”

The winner beat one opponent by 22 votes (x – 22), another by 30 votes (x – 30), and the last by 73 votes (x – 73).

We can build our simple equation from that information:

x + (x – 22) + (x – 30) + (x – 73) = 5219

Still a little daunting, but we can simplify it, because it doesn’t matter in which order we add or subtract things. So let’s look at that formula without the parentheses:

x + x – 22 + x – 30 + x – 73 = 5219

Now let’s reorganize it, putting the addition parts together and the subtraction parts together:

x + x + x + x – 22 – 30 – 73 = 5219

Subtracting those three numbers separately is the same as subtracting their total, so let’s simplify again:

x + x + x + x – 125 = 5219

Adding four x’s together is the same as multiplying one x by 4, so let’s express that:

4x – 125 = 5219

Now we’re getting somewhere.

And subtracting 125 from 4x is the same as adding 125 to 5219, so let’s do that:

4x = 5344

Finally, we divide 5344 by 4 to give us the value of x:

x = 1336

Which means that our victor got 1336 votes, one opponent got 1314 (x – 22), another opponent got 1306 (x – 30), and the last got 1263 (x – 73), totalling 5129 votes.

Now, that wasn’t so bad, was it? Let’s try another that’s a little bit harder.

This one is called “The Mathematical Cop.”

“Top of the mornin’ to you, officer,” said Mr. McGuire. “Can you tell me what time it is?”

“I can do that same,” replied Officer Clancy, who was known on the force as the mathematical cop. “Just add one quarter of the time from midnight until now to half the time from now until midnight, and it will give you the correct time.”

Can you figure out the exact time when this puzzling conversation took place?

Okay, this one isn’t as obvious about providing us with information, but the info is there if you look.

Since everything relates to the time “now,” we’ll make “now” our x.

Then we take each part of Officer Clancy’s statement in turn. “Just add one quarter of the time from midnight until now.”

“The time from midnight until now” is the same as “now,” x, so one quarter of that time is x/4.

And we’re meant to add that to “half the time from now until midnight.”

That’s a little bit tougher. After all, “the time from midnight to now” was easy, but “the time from now until midnight” covers the rest of a 24-hour day. So, if x covers the time from midnight to now, then “1440 – x” covers the time from now until midnight.

(There are 1440 minutes in a day, 60 minutes times 24 hours, and it’s easier to do all this in minutes, rather than hours and minutes.)

So “half the time from now until midnight” is (1440 – x)/2.

Okay, so what does our equation look like?

x/4 + (1440 – x)/2 = x

That’s pretty daunting, but we know what our goal is: to combine all those x’s and get them on the same side of the equal sign. And like the equation we built for “Count the Votes,” we can simplify it with some careful applied math.

The first step is to get rid of those pesky fractions.

Let’s multiply everything by 2 in order to remove the “/2” below “(1440 – x),” which gives us:

2x/4 + (1440 – x) = 2x

We can use the same trick to remove the “/4” below 2x:

2x + 4(1440 – x) = 8x

Now we’re getting somewhere! Let’s get rid of that 2x on the left by subtracting 2x from both sides:

4(1440 – x) = 6x

Let’s go a step further by multiplying both 1440 and x by 4:

5760 – 4x = 6x

One more step, and we’ve got all of those x’s combined on one side of the equation, as we’d hoped:

5760 = 10x

Divide 5760 by 10 and we’ve got x:

576 = x

If you recall, x represented the time “now,” but it’s still in minutes. To get the actual time, divide 576 by 60 to get the number of hours. 540 minutes = 9 hours, so 576 is 9 hours, 36 minutes.

It’s 9:36 AM, Officer, though to be honest, if you tell everyone the time this way, I imagine people stop asking you the time after a while.

I realize these are only two examples, and math puzzles come in all shapes and sizes, but hopefully, they don’t seem quite so intimidating, now that you know how to pick them apart for the important information.

Good luck! And if you find any math puzzles you need help with, send them our way! They could end up the subject of a future blog post!

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Cracking the GCHQ Christmas Card!

As you may recall, my fellow puzzlers and PuzzleNationers, a few months ago, a government organization in England called the GCHQ — Government Communications Headquarters — released a puzzly Christmas card designed to tax even the savviest puzzle solvers.

They’ve finally released the answers to this mind-blowing series of puzzles, and I’d like to go over some of them with you. Partly to marvel at the puzzle wizardry necessary to solve this challenging holiday gift, and partly to gloat about the parts I managed to solve.

So let’s get to it!

Part 1 was a logic art puzzle where you have to deduce where to place black squares on an open grid in order to form a picture.

Each column and row has a series of numbers in it. These numbers represent runs of black squares in a row, so a 1 means there’s one black square followed by a blank square on either side and a 7 means 7 black squares together with a blank square on either side.

This is mostly a deduction puzzle — figuring out how to place all the strings of black squares with white spaces between them within the space allotted — but no image immediately emerged, which was frustrating. Once the three corner squares started to form though, I realized the answer was a QR code, and the puzzle started to come together nicely.

Part 2 was a series of six multiple-choice brain teasers. I’ll give you the first three questions, along with answers.

Q1. Which of these is not the odd one out?


Now, if you stare at a list of words long enough, you can form your own patterns easily. Here’s the rationale the GCHQ used to eliminate the odd ones out:

STARLET is an odd one out because it does not contain a double letter.
SONNET is an odd one out because it has 6 letters rather than 7.
SAFFRON is an odd one out because it ends in N rather than T.
TORRENT is an odd one out because it starts with T rather than S.
SUGGEST is an odd one out because it is a verb rather than a noun.

SHALLOT is our answer.

Q2. What comes after GREEN, RED, BROWN, RED, BLUE, -, YELLOW, PINK?

After playing around with some associative patterns for a while, I realized that somehow these colors must equate to numbers. First I tried word lengths, but 5-3-5-3-4-___-6-4 didn’t make any sense to me. But then, it hit me: another time where colors and numbers mix.

Pool balls. Of course, the colors and numbers didn’t match, because this is a British puzzle, and they don’t play pool, they play snooker.

So the colored balls in snooker become the numbers 3, 1, 4, 1, 5, -, 2, 6. The numbers of Pi. And now the blank makes sense, because Pi reads 3.1415926, and there’s no 9 ball in snooker.

So the next number in the chain is 5, and 5 is the color BLUE.

Q3. Which is the odd one out?

This one came pretty quickly to me, as the names Oscar Wilde and Charlie Bucket leapt out. And if you follow the phonetic alphabet, you also get Victor Mature, Romeo Beckham, and Shaka Zulu. (I didn’t get Mike Love, however.)

Since Shaka Zulu was the only one where the phonetic alphabet word was the surname, not the first name, SHAKA is the odd one out.

(The other three questions included an encryption puzzle, a number pattern (or progressions puzzle), and a single-letter puzzle.)

Granted, since you could retake this part as many times as you wanted, you could luck your way through or brute force the game by trying every permutation. But managing to solve most of them made this part go much faster.

Part 3 consisted of word puzzles, and was easily my favorite section, because it played to some strengths of mine.

A. Complete the sequence:

Buck, Cod, Dahlia, Rook, Cuckoo, Rail, Haddock, ?

This sequence is a palindrome, so the missing word is CUB.

B. Sum:

pest + √(unfixed – riots) = ?

This one is a little more involved. To complete the formula, you need to figure out what numbers the words represent. And each word is an anagram of a French number. Which gives you:

sept + √(dix-neuf – trois) = ?

Dix-neuf is nineteen and trois is three, so that’s sixteen beneath a square root sign, which equals four. And sept (seven) plus four is eleven.

The French word for eleven is onze, and ZONE is the only anagram word that fits.

C. Samuel says: if agony is the opposite of denial, and witty is the opposite of tepid, then what is the opposite of smart?

This is a terrific brain teaser, because at first blush, it reads like nonsense, until suddenly it clicks. Samuel is Samuel Morse, so you need to use Morse Code to solve this one. I translated “agony” and tried reversing the pattern of dots and dashes, but that didn’t work.

As it turns out, you need to swap the dots and dashes, and that’s what makes “denial” read out. This also worked with “witty” and “tepid,” so when I tried it with “smart,” the opposite was OFTEN.

D. The answers to the following cryptic crossword clues are all words of the same length. We have provided the first four clues only. What is the seventh and last answer?

1. Withdraw as sailors hold festive sing-song
2. It receives a worker and returns a queen
3. Try and sing medley of violin parts
4. Fit for capture
7. ?

Now, I’m not a strong cryptic crossword solver, so this part took FOREVER. Let’s work through it one clue at a time.

1. Withdraw as sailors hold festive sing-song

The word WASSAIL both reads out in “withdraw as sailors hold” and means “festive sing-song.”

2. It receives a worker and returns a queen

The word ANTENNA both “receives” and is formed by “a worker” (ANT) and “returns a queen” (ANNE, reading backward).

3. Try and sing medley of violin parts

The word STRINGY is both an anagram of “try” and “sing” and a violin part (STRING).

4. Fit for capture

The word SEIZURE means both “fit” and “capture.”

Those four answers read out like this:


And with three more answers to go, it seemed only natural that three more seven-letter answers were forthcoming. Plus, when you read the words spelling out downward, you notice that the first four letters of WASSAIL, ANTENNA, STRINGY, and SEIZURE were spelling out.

If you follow that thought, you end up with the start of a 7×7 word square:


And the only seven-letter word starting with INGR that I could think of was INGRATE.


And if the last word is LAYERED…


Then the missing word must be ANNULAR. The original question asked for the last word though, so our answer is LAYERED.

This brings us to Part 4, Number Puzzles, where I must confess that I finally tapped out, because I could only figure out the first of the three progressions involved.

Fill in the missing numbers.

A. 2, 4, 8, 1, 3, 6, 18, 26, ?, 12, 24, 49, 89, 134, 378, 656, 117, 224, 548, 1456, 2912, 4934, 8868, 1771, 3543, …

B. -101250000, -1728000, -4900, 360, 675, 200, ?, …

C. 321, 444, 675, 680, 370, 268, 949, 206, 851, ?, …

In the first one, you’re simply multiplying by 2 as you go.

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, and so on.

But you begin to exclude every other number as you move into double-digits, triple-digits, quadruple-digits, and beyond.

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, and so on.

So the answer, 512, becomes the real answer, 52.

But, as I said, I couldn’t crack the other two, and I’m already exhausted just running through these four sections!

And, based on the answers they released recently, Part 5 only got more mindbending from there.

As a matter of fact, not a single entrant managed to get every answer in Part 5 correct. Prizes were awarded to the three people who came closest however, and it turns out a staggering 30,000+ people made it to Part 5. Color me impressed!

This was, without a doubt, the most challenging puzzle suite I have ever seen, and I offer heartfelt kudos to anyone in the PuzzleNation Blog readership who even attempted it!

You’re welcome to try it out for yourself, though. I highly recommend using this link from The Telegraph, which allows you to skip to the next part if you get stumped.

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