Set to release at ETHDenver 2023, logicbeach.eth announced on his twitter feed on the 24th of February that Supermodular.xyz’s book would contain another one of his great puzzles on it’s cover !
This would be the 3rd puzzle that I have tried and luckily, even though I couldn’t go to Denver, he would publish everything needed to solve it on his twitter feed.
The puzzle would come with a prizepool in the form of a wallet, funded partly by Logicbeach, partly by the sales of a POAP (holders were eligible to enter a raffle to win a copy of the book and a special ETHDenver LogicBot). He would also include an amazing NFT from his LogicBots collection (drawing one robot a day, everyday).
https://twitter.com/Logic_Beach/status/1629203055165034499
You can find the book for purchase here :
https://www.blurb.fr/b/11476889-stuff-crypto-ogs-know
The first tweet indicated that the private key would be hidden in plain sight on the front cover of the book but a hint would be needed to solve the puzzle.
Below is the first hint given during the ETHDenver book launch on the 2nd of March.
After studying the hint, I could spot four different steps, or puzzles, to solve :
polish spiral
the integral
“after you know” part
the background picture
Ok, I lack quite alot of the more advanced mathematical notions, having stopped at highschool, so thought I would ask ChatGPT what a ‘polish spiral’ could be. It’s answer led me down a bad path for quite a while…
It took me ages to realize that an Archimedean spiral was not what I was looking for, which led me to Google (no comment… :D). Searching ‘polish spiral’ did not get me anywhere though and it was only when including ‘maths’ in the search terms (yes maths, not math) did I see ‘Stanislaw Ulam - Wikipedia’ which suddenly reminded me of one of the comments I had seen on the twitter feed citing Ulam ! (thanks to @lahcen_kha)
Stanislaw Ulam was a Polish scientist in the fields of mathematics and nuclear physics and the wikipedia page had a picture that was exactly (or nearly, we will see later…) the same one as provided in the hint. I had to be on the correct route now !
Ulam’s spiral is a graphical depiction of the set of prime numbers that is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.
After a search online for an advanced calculater, I landed on WolframAlpha (that I had seen in use before on Numberphile’s youtube videos). Using their natural language calculator, I entered : “the integral of x exponential of -x squared from minus infinity to positive infinity“ and this was the result : 0
With this I knew that Ulam’s spiral had to start on the first 0 “after you know”.
This is where things went wrong. I hoped that ChatGPT might be able to solve something like this, so I entered a table with all the characters featured on the book’s cover. And ChatGPT managed to solve this (or so I thought) and returned a paragraph that actually made sense :
I thought to myself : ‘I now KNOW’. The above started with the word KNOWLEDGE so Ulam’s spiral must be done on this paragraph and not on the book’s cover. Converting this to hexadecimal gave me a new table of characters to play with. The problem was where to start the spiral now as there were lots of 0s in this new table.
I tried loads of things : starting the spiral on the first 0, trying to find the characters that corresponded to the first 0 after KNOW on the book’s cover etc Nothing seemed to work though.
I knew I needed to find an ethereum private key and so continued my research on these keys and their formats. It needed to be a 64 length hexadecimal value.
When I finally got to the limit of my ChatGPT requests I moved over to Photoshop. You might be wondering “why photoshop ?” Well I had found a Ulam’s spiral generator online
https://www.geogebra.org/m/Q9Hk4Uyn
and thought that by superposing this with the characters in my new table I might make some progress. I did not… All these tries produced more or less 20 private keys that were all imported into Metamask (ethereum wallet), sadly all of them were empty !
This new table of hexadecimal values converted from the paragraph that ChatGPT had returned just didn’t seem to be doing the job.
After starting fresh, I then thought what if ChatGPT had returned something completely different and tried then to apply Ulam’s Spiral directly on the book’s cover, starting at the first 0 after KNOW.
As you can see, once I had mucked around with the size of Ulam’s Spiral and it’s transparency, I could follow it’s path and collect the values corresponding to the little red dots, aka the position of the prime numbers. Here goes 300580……
This led to another problem (and quite a few more empty wallets imported to MetaMask). Some prime positions corresponded to values that were not hexadecimal (between 0-9 and A-F). I did not know whether to just leave them out or convert them to hexadecimal. Well I finally just tried them all but still nothing was working. Bed time !
As mentionned previously, the background picture of the first hint was Stanislaw Ulam. It had been taken from wikipedia. This is where Logigbeach’s last hint came in real handy : “There is one more hint in the image…. it’s right in front of you and not in words”
The only thing I could see was that it was a ‘negative’ of the original wikipedia image. And when solving any puzzle, when new possible solutions come up the hype is back on !!! Searching for ‘negative Ulam spiral’ on google did not return much. And by the usual definition of prime for integers, negative integers can not be prime. I tried Ulam’s spiral backwards, I tried vertical/horizontal symmetry all leading to many more hexadecimal private keys and more empty wallets.
Then… yes this is that EUREKA moment, what if… should I make a negative copy of Ulam’s spiral and superpose that on the book’s cover ? All the dots would invert and instead of the primes being shown it would display the non-primes.
All I needed to do was follow the spiral, any time there was a black dot it was a non-prime and after noting those down I still had a few options. Leave the non-hexadecimal values out (‘T’ and ‘O’ from the end of CRYPTO word) or convert them to hexadecimal, respectively 54 and 4F.
They finally did have to be converted as they led to my 45th wallet import which contained the prizepool, NFTs but also and mostly led to an extreme feeling of accomplishment and joy !
As mentionned, I had previously participated in Logicbeach’s puzzles and got quite close, but this time managed to hit the robot’s eye !
A big thanks to logicbeach.eth for this puzzle, very entertaining and just complex enough that it took 3 days to solve (probably around 5 hours) but not too difficult so that I could decipher it :)