While in Southern France, Simon really enjoyed solving this puzzle (he originally saw in a Brilliant.org vid). He was so happy with his solution he kept drawing it out on paper and in digital apps, and later shared the puzzle on Twitter. This sparked quite a few reactions from fellow math lovers, encouraged Brilliant to tweet new puzzles and now Brilliant follows Simon on Twitter, how cool is that!
Simon teaching his sister Neva from the Mathematical Fundamentals course on Brilliant:
Had great fun learning how to crack codes using Python! Simon is currently following the Programming with Python course on Brilliant.org and showed me how to see whether an encrypted piece is gibberish or a real text is hidden behind it.
A Caesar Shift is a simple cipher, which was a standard in Roman times. It works like this: shift every character by some fixed amount in the alphabet. Something like this:
Example: Suppose some professor writes his name on his board:
It’s encoded with a caesar shift. Because it’s a professor’s name, it probably starts with “Dr.”, so it’s probably a shift that turns D into E, and R into S. So we can work backwards from that shift, and get:
That was an easy one, so let’s do something more complex with code.
One of the messages below is a real text, encoded using a Caesar Shift, the other one is just a random sequence of letters. Can you tell which one is which?Text 1:
yfdpcpoplhhwdpssbjnsqvtlcpzpxqugtjphvgotuvwxufgoqigxwgkskduooyeuoue fjlnmsqpgxrmcseeliswdheywseqgcbeothskxdzekgxmmkildjnaqbukprpfaaknsu qpdwayqaqfxsoapvsgreqydqjnkpjghvrkygtidzibhrqkmocukhcunpjcazzvomtsc fgycwfltmiegaejwcqrgsnxxcbtcrckufwsdxdhbxgppxcuzapbdhftzmugryfseavv bssqlxanvmfwwzityziixasivzkmvtfczqmdgkabcnjbyhaoealengfptuedlmvryeb titbwqkekzdpmbtiphdkwwiduassvbgalxgrfhrjrjplxpujrprqzcpcdqsjorigazt kwwlnwbjryrzhgcttroyemuwwixwufymnknirzmexyowobvardlqktzajzoijwulomg ztefdpftjealzapcgipgaaspuzxklvdText 2:
swodkdbkfovvobpbywkxkxdsaeovkxngrycksndgyfkcdkxndbexuvoccvoqcypcdyx ocdkxnsxdronocobdxokbdrowyxdrockxnrkvpcexukcrkddobonfsckqovsocgrycop bygxkxngbsxuvonvszkxncxoobypmyvnmywwkxndovvdrkdsdccmevzdybgovvdrycoz kccsyxcbokngrsmriodcebfsfocdkwzonyxdrocovspovoccdrsxqcdrorkxndrkdwym uondrowkxndrorokbddrkdponkxnyxdrozonocdkvdrocogybnckzzokbwixkwoscyji wkxnskcusxqypusxqcvyyuyxwigybuciowsqrdikxnnoczksbxydrsxqlocsnobowksx cbyexndronomkiypdrkdmyvycckvgbomulyexnvocckxnlkbodrovyxokxnvofovckxn ccdbodmrpkbkgki
Simon has explained a way to see whether the encrypted piece contains meaningful (real) text: one can plot the frequency of each letter as it’s used in the encrypted piece. If all letters have generally similar frequency, it’s not a real text, because in real texts, certain letters are encountered much more often than others. Below are the frequency plots Simon made for the texts above, using a Python package called
Frequencies for text 1:
Frequencies for text 2:
As you can see, the second plot depicts a greater variety in frequencies. “For example, o appears the most, but g does not appear that much. And t does not appear at all!” Simon showed me.
As it turned out, we could actually use our knowledge about which letters naturally appear more frequently in English-language texts to crack the code! “Which letter is the most frequent one in English writing?” Simon asked me. “Letter e!” I guessed. “So now we know that the letter o in the encrypted text stands for e in the real text!” Simon exclaimed. “All we have to do to decode it now is simply shift the letters by 10 letters back, because e is 10 letters behind the o!”
So, what is the message about? Simon tweaked Brilliant’s code to make sure it shifted by the amount of 10…
…put the spaces and punctuation in appropriately…
I met a traveller from an antique land
Who said: “Two vast and trunkless legs of stone
Stand in the desert . . . Near them, on the sand,
Half sunk, a shattered visage lies, whose frown,
And wrinkled lip, and sneer of cold command,
Tell that its sculptor well those passions read
Which yet survive, stamped on these lifeless things,
The hand that mocked them, and the heart that fed:
And on the pedestal these words appear:
‘My name is Ozymandias, king of kings:
Look on my works, ye Mighty, and despair!’
Nothing beside remains. Round the decay
Of that colossal wreck, boundless and bare
The lone and level sands stretch far away.”
So, it’s about Archeology! This is the poem Ozymandias by Percy Shelley (1818).
Encoder / Decoder:
alphabet = "abcdefghijklmnopqrstuvwxyz" # convert between letters and numbers up to 26 def number_to_letter(i): return alphabet[i%26] def letter_to_number(l): return alphabet.find(l) # How to encode a single character (letter or not) def caesar_shift_single_character(l, amount): i = letter_to_number(l) if i == -1: # character not found in alphabet: return "" # remove it, it's spaces or punctuation else: return number_to_letter(i + amount) # Caesar shift # How to encode a full text def caesar_shift(text, amount): shifted_text = "" for char in text.lower(): # also convert uppercase letters to lowercase shifted_text += caesar_shift_single_character(char, amount) return shifted_text ### MAIN PROGRAM ### message = """ paste the text here """ code = caesar_shift(message, 2) print(code)
Code for Plots:
import matplotlib.pyplot as plt alphabet = "abcdefghijklmnopqrstuvwxyz" code = """ paste the text here """ letter_counts = [code.count(l) for l in alphabet] letter_colors = plt.cm.hsv([0.8*i/max(letter_counts) for i in letter_counts]) plt.bar(range(26), letter_counts, color=letter_colors) plt.xticks(range(26), alphabet) # letter labels on x-axis plt.tick_params(axis="x", bottom=False) # no ticks, only labels on x-axis plt.title("Frequency of each letter") plt.savefig("output.png")
Simon loves challenging other people with math problems. Most often it’s his younger sister Neva who gets served a new portion of colourful riddles, but guests visiting our home also get their share, as do Simon’s Russian grandparents via FaceTime. Simon picks many of his teaching materials in the Mathematical Fundamentals course on Brilliant.org, and now Neva actually associates “fundamentals” with “fun”!
Sunday at the beach, Simon was reenacting the 5 doors and a cat puzzle (he had learned this puzzle from the Mind Your Decisions channel). The puzzle is about guessing behind which door the cat is hiding in as few guesses as possible, while the cat is allowed to move one door further after every wrong guess.
“Here’s a fun fact!” Simon said all of a sudden. “If you add up all the grains of sand on all the beaches all over the world, you are going to get several quintillion sand grains or several times 10^18!” He then proceeded to try to calculate how many sand grains there might be at the beach around us…
In the evening, while having a meal by the sea, Simon challenged Dad with a Brilliant.org problem he particularly liked:
Simon’s explanation sheet (The general formulas are written by Simon, the numbers underneath the table are his Dad’s, who just couldn’t believe Simon’s counterintuitive solution at first and wanted check the concrete sums. He later accepted his defeat):
Simon finds the explanation on Brilliant.org incomplete, so he started a discussion about it on the Brilliant community page: https://brilliant.org/discussions/thread/games-of-chance-course-marble-problem/?ref_id=1570424
Finally, parts 6 and 7 of Simon’s exciting series of video tutorials about sorting algorithms are done! In the videos, Simon codes on his RaspberryPi, but here is the link to the Python code (parts 6 – 7) available on his GitHub page:
The code of the sorting algorithms discussed in the previous videos (parts 1 – 5) is available here: https://gist.github.com/simon-tiger/5be70247a066f69c2578be5bb8e41e59
Simon wrote the Shellsort code himself. He tried to run his own code for Heapsort as well, but didn’t get the list fully sorted, so in the end he implemented the heapsort code that he learned from Brilliant.
“Then, with VERY much relief, I MASSIVELY condensed the code (to just 3 lines!), using Zax Rosenberg’s blog“, Simon adds.
Simon writes (this is a copy of the question he has published in Coding Train’s Slack chat):
Hi! I was unconvinced on a Brilliant puzzle:
The problem (at https://brilliant.org/practice/treaps/?p=4):
Me working on it…
The solution on Brilliant was different:
However, only 1 page earlier, the explanation said:
“Mom, how long would it take a supercomputer running at 10^15 additions per second to calculate the 1000th Fibonacci number?”
Simon has learned this problem from the new course he is following on Brilliant.org: Computer Science Algorithms. Simon worked it out on an A3 sketch book sheet and got the answer correct: it would take longer than the age of the Universe!
Simon has already finished the Computer Science Fundamentals course! It has been Simon’s idea to take up the courses on Brilliant.org again and he has been working independently, driven entirely by his intrinsic motivation.
The course has also inspired Simon to work on a very large scale project: record a series of tutorials where he explains all the best known sorting algorithms and comes up with the Python code for them on his RaspberryPi!
From Yesterday’s math class. Simon continued with the equations and managed to complete four quizzes on Brilliant.org within one hour. He neglects writing things down though, trying to solve everything in his mind. So far it has been a success, but both his math tutor and I are trying to explain it to him that the more complex equations are easier solved when writing them out. Algebra > Solving Equations > Setting Up Equations, Simple Equations, Multi-Step Equations, Isolating a Variable.