Happy 6/7 Day

6/7 is here!
Most of Earth got the date right;
Americans cope.

Opening words

I opened with a haiku because why not? Unless you’ve been living under a rock for the past 6-7 months, you will likely have encountered 6 7. I mean it’s even Dictionary.com’s 2025 Word of the Year.

But also, this post doubles up as a personal reflection for SekaiCTF 2026 which ended 6-7 days ago, for which I wrote three challenges.

Challenge	Category	Difficulty	Solves
6-7 Puzzle Hunt	Game	Easy	362
oneline6ryp7o	Crypto	Easy	423
apbq-rsa-iv	Crypto	Hard	11

That last one is not 6/7-related and has been split out into its own post.

SekaiCTF 2026

It’s quite possible that this might be the final SekaiCTF, at least in its current jeopardy format. I’ve contributed at least 3 challenges in each of 2024, 2025, 2026, but I think this last one’s been the hardest to set the difficulty right. In the past I just wrote the hardest possible challenges, but this time LLMs have just gotten too good, so I went into this with a different approach.

Namely, I aimed for every challenge to be either unsloppable (too hard for AI) or fun-and-solvable by hand. Or by hands.

6-7 Puzzle Hunt

If you haven’t attempted this, do try it out here! It’s a collection of six (or seven) pure logic puzzles, all seeded with only 6s and 7s.

Solution (spoilers – click to expand)

Putting the jigsaw pieces together gives you

┏━━━━━━━━━━━━━━━┳━━━┳━━━━━━━┓
┃ S   E   K   A ┃ I ┃ {   S ┃
┣━━━━━━━┓   ┏━━━┛   ┗━━━┓   ┃
┃ C   R ┃ I ┃ B   B   L ┃ E ┃
┃   ┏━━━┛   ┃   ╺━━━┓   ┃   ┃
┃ _ ┃ 6   7 ┃ _   L ┃ I ┃ K ┃
┃   ┣━━━┳━━━┻━━━━━━━╋━━━┫   ┃
┃ E ┃ _ ┃ I   T   S ┃ _ ┃ A ┃
┃   ╹   ┃   ┏━━━┳━━━┫   ┃   ┃
┃ _   S ┃ E ┃ X ┃ Y ┃ _ ┃ I ┃
┣━━━━━━━┛   ┃   ╹   ╹   ┃   ┃
┃ N   T   E ┃ G   E   R ┃ } ┃
┗━━━━━━━━━━━┻━━━━━━━━━━━┻━━━┛

for the flag SEKAI{SCRIBBLE_67_LIKE_ITS_A_SEXY_INTEGER}.

Your most important takeaway from this puzzle should be that 67 is definitely sexy.

This puzzle went through a lot of iterations actually. Things that were originally considered:

• Wordplay puzzles: Think crosswords and stuff. But I eventually decided this would not be fair for non-English speaking teams.
• Grids that were not 10x10: In particular, slitherlink is a favourite of mine but whose numbers only go up to 4, but @hellman and @soon_haari noted that it can be played on an octagonal grid.
• Puzzles types not on puzz.link: Having all the puzzles on puzz.link means I didn’t need to write much custom code, as it already did all the checking and undoing and stuff.

Fillomino Lemma

@chsh noted that the fillomino didn’t have the usual Neobeo elegance because there were regions that were not of sizes 6 or 7, so here’s a quick proof that this is not possible.

A 6-region must have perimeter in [10,12,14]. A 7-region must have perimeter in [12,14,16]. If every region has area 6 or 7, there are two cases:

• 12×6 + 4×7: The sixes have perimeter ≥120 and the sevens have perimeter ≤64.
• 5×6 + 10×7: The sixes have perimeter ≤70 and the sevens have perimeter ≥120.

In either case, the perimeter difference exceeds the excess perimeter of 40 for a 10x10 grid, thus completing the proof.

Favicon

Also, I’m kinda proud of the 100% hand-drawn favicon. Give me some credit.

oneline6ryp7o

Here’s the entire challenge. Yes it’s a one-liner.

assert __import__('re').match('SEKAI{[67]{67}}$',flag:=input()) and not int.from_bytes(flag.encode())%~(6+~7)**67

This is a variant of one of my most widely cited challenges in the past, and in fact is one character shorter than the original (114 → 113)!

I wrote onelinecrypto for SEETF 2023, and in my writeup I made this insignificant comment:

It turns out my original choice of $n = 2^{255}-19$ was not a good idea, for reasons I will not explain in case I manage to turn it into a different challenge.

And this is that challenge! Though I went with $2^{67}-1$ instead, but the idea still stands. Also worth mention that writing ~(6+~7)**67 is just an aesthetic thing so that the only digits in the line are “67676767”.

Fun fact about $2^{67}-1$: Mersenne originally claimed in 1644 that it was prime. It wasn’t until 1903 that its factorisation was known, with Frank Nelson Cole famously giving a silent lecture, literally just multiplying 193707721 × 61838257287 on a blackboard. And then everyone clapped.

But anyway, using a modulus near an odd power of 2 allows a bunch of shortcuts. This is because

\[2^k \equiv 2^{k-67} \pmod{2^{67}-1},\]

and since $256=2^8$ and $\gcd(8,67)=1$, we can reduce anything to just 67 consecutive bits this way.

Solve by hand

I did say the 67 challenges were designed to be solved by hand, so here goes. Though you do need to have an ASCII table handy somewhere.

S = 0x53 = 01010011
E = 0x45 = 01000101
K = 0x4B = 01001011
A = 0x41 = 01000001
I = 0x49 = 01001001
{ = 0x7B = 01111011
} = 0x7D = 01111101

You can concatenate them into a string of 56 bits, and then pre-pad them with another eleven 0s, so it’s length 67. Then replace each digit $d$ with $7-d$:

0000000000001010011010001010100101101000001010010010111101101111101
7777777777776767766767776767677676676777776767767767666676676666676

Finally read every 8th digit starting from the last digit and wrapping round. So something like this, reading down the columns in the numbered order:

63852741
========
77777777
77776767
76676777
67676776
76676777
77676776
77676666
76676666
676
========
  ^-- start from this last digit and go down

The flag is thus SEKAI{6777676667666666677676776776777766777777777776777767777776677666666}.

one line solution

print('SEKAI{%s}'%''.join('76'[int(b'SEKAI{}'.hex(),16)>>-i*8%67&1]for i in range(67)))

apbq-rsa-iv

Ok, I did say this is not 6/7-related, but also the flag is actually the following cryptic clue:

(Fine, I had to inspect-element-edit this myself, but still a shoutout to minutecryptic.com)

I’ve never heard it in daily use, but “at sixes and sevens” is an English idiom used to describe a condition of confusion or disarray. So it’s an anagrind for RURAL AMBIANCE for the final answer ARABIC NUMERAL. “6 or 7” being the definition.

Other 67 fun stuff

Wordle party trick

This is a fun party trick, and I use to annoy my daughter. Basically I pretend to make some terrible guesses, and at the end of the game I will have painted 67 somehow.

She never lets me make the first guess anymore.

Anyway, it also looks great on discord:

So I vibecoded a thing to let people do this: Wordle Painter

6x7 crossword grid

I also attempted to make a 6x7 crossword grid. With PROJECT as one of the entries, because you know, PROJECT SEKAI. I ended up with a bunch of spurious words, but it’s not the worst grid:

M A S S O R A
A R T E M I S
P R O J E C T
P I R A N H A
E V E N T E R
R E S T A R T

Heegner numbers

There are exactly nine Heegner numbers, the largest two of which are 67 and 163. And I wrote a challenge on exactly this for SekaiCTF 2024 called √163, so it’s nice that it can be done with 67 as well.

Anyway, the main fun trick is the near-integer

\[e^{\pi \sqrt{67}} \approx 5280^3 + 744 - 0.0000013.\]

Closing remarks

Since I opened with a haiku, guess I’ll close with a limerick.

There once was a man from Devon
Who gathered fun facts on six-seven
He said “It’s been great
But today is the date
We lay it to rest in meme heaven.”