(I wrote this piece for La Recherche. It was translated into French by Philippe PAJOT. You can find this piece and pieces by others at John Horton Conway: a magician of maths disappears.)
Unlike many other mathematicians I know, John Conway cared a lot about the way he presented things. For example, in the puzzle he invented—known as Conway’s Wizards—the wizards had to be riding on a bus. Why was the bus so important? You see, the numbers in the puzzle were related to the age of one of the wizards, the number of the bus, and the number of the wizard’s children. It was important to John that the readers be able to use a convenient notation a, b and c for these numbers and remember which number is which.
When I give my lecture on integers and sequences, I show my students a list of different famous sequences. The first question from the audience is almost always: “What are the Evil Numbers?” As you can guess the name for this sequence was invented by John Conway. This name was invented together with the name of another sequence which is called Odious Numbers. These two sequences are complementary in the same sense as even and odd numbers are complementary: every natural number is either evil or odious. The names are good, not only because they attract, but also because they help remember what the sequences are. Evil numbers are numbers with an even number of ones in their binary representation. I assume that you can interpolate what the odious numbers are.
When he was lecturing, John used all sorts of tricks to emphasize important points: From time to time I saw him shouting or throwing his shoes. Once I remember him staring at his statement written on the blackboard for a really long time. My neighbor in the lecture hall got uncomfortable. He assumed that John, who was at that time way over 70, was blanking out and had forgotten what he wanted to say. I calmed my neighbor down. It was my fourth time listening to the same lecture, including the same pause. John Conway didn’t forget.
Every year I review MIT mystery hunt from a mathematician’s point of view. I am way behind. The year is 2020, but I still didn’t post my review of 2019 hunt. Here we go.
Every year I review MIT mystery hunt from a mathematician’s point of
view. I am way behind. The year is 2020, but I still didn’t post my
review of 2019 hunt. Here we go.
Many puzzles in 2019 used two data sets. Here is the recipe for
constructing such a puzzle. Pick two of your favorite topics: Star Trek
and ice cream flavors. Remember that Deanna Troi loves chocolate sundae.
Incorporate Deanna Troi into your puzzle to justify the use of two data
On one hand, two data sets guarantee that the puzzle is new and fresh.
On the other hand, often the connection between two topics was forced.
Not to mention that puzzle solving dynamic is suboptimal. For example,
you start working on a puzzle because you recognize Star Trek. But then
you have to deal with ice cream which you hate. Nonetheless, you are
already invested in the puzzle so you finish it, enjoying only one half
Overall, it was a great hunt. But the reason I love the MIT mystery hunt
is because there are a lot of advanced sciency puzzles that can only
appear there. For example, there was a puzzle on Feynman diagrams, or on
characters of representations. This year only one puzzle, Deeply Confused, felt like AHA, this is the MIT Mystery hunt.
Before discussing mathy puzzles I have to mention that my team laughed at Uncommon Bonds.
I will group the puzzles into categories, where the categories are obvious.
Here are some logic puzzles, in a sense that Sudoku is a logic puzzle.
- Lantern festival—A cool mixture of Slitherlinks and Galaxies.
- Invisible Walls.
- Place Settings.
- Middle School of Mines—Minesweeper.
- Moral Ambiguity—Nonograms with a twist.
- Connect Four—Mastermind.
There was a strong hint that the extraction step was also mastermind.
My team spent some time trying to mastermind the ending, until we
backsolved. The extraction step was not mastermind. The final grid in
the puzzle had the word CODE written in red. It corresponded to letters
CDEO found at that location. Given that the letters were not in
alphabetical order, it gave the ordering, which didn’t exist in the
puzzle. Anyway, you can see that I have a grudge against this puzzle.
This could have been a great puzzle. But it wasn’t.
- Schematics—Tons of Nikoli puzzles of different types.
Now we have logic puzzles or another type, where you need to draw a grid. These are puzzles of the type: Who lives in the White House?
Now we have logic puzzles or yet another type, where you need to figure out which statements are true and which are false.
Now some cryptography.
And some programming.
Every day I check coronavirus numbers in the US.
Right now the number of deaths is 288 and the number of recovered is
171. More people died than recovered. If you are scared about the
mortality rate, I can calm you and myself down: our government is
incompetent—the testing wasn’t happening—that means the numbers do not
show people who had mild symptoms and recovered. The real number of
recovered people should be much higher.
Scientists estimated the mortality rate of coronavirus as being between 1
and 3.5 percent. Also, they say that it usually takes three weeks to
die. That means three weeks ago the number of infected people in the US
was between 8,000 and 29,000. The official number of cases three weeks
ago was 68. I am panicking again—our government is incompetent—three
weeks ago they detected between 0.25 and 1 percent of coronavirus cases.
If this trend continues, then the official 19,383 infected people as of
today means, in reality, somewhere between 2 million and 8 million
I can calm you and myself down: the testing picked up pace. This means,
the ratio of detected cases should be more than 1 percent today.
Probably the number of infected people today in the US is much less than
8 million. I am not calm.
My friend Zeb, aka Zarathustra Brady,
invented a new game that uses chess pieces and a chessboard. Before the
game, the players put all chess pieces on white squares of the board:
white pieces are placed in odd-numbered rows and black pieces are in
even-numbered rows. At the beginning all white squares are occupied and
all black squares are empty. As usual white starts.
On your turn, you can move your piece from any square to any empty
square as long as the number of enemy neighbors doesn’t decrease. The
neighbors are defined as sharing a side of a square. Before the game
starts each piece has zero enemy neighbors and each empty square has at
least one white and one black neighbor. That means that on the first
turn the white piece you move will increase the number of neighbors from
zero to something.
As usual, the player who doesn’t have a move loses.
As you can immediately see, that number of pairs of enemy neighbors is
not decreasing through the game. I tried to play this game making a move
which minimizes the increase of the pairs of neighbors. I lost, twice. I
wonder if there is a simple strategy that is helpful.
It is important that this game is played with chess pieces in order to
confuse your friends who pass by. You can see how much time it takes
them to figure out that this game is not chess, but rather a Chessnot.
Or you can enjoy yourself when they start giving you chess advice before
realizing that this is not chess, but rather a Chessnot.
I heard this puzzle many years ago, and do not remember the origins of it. The version below is from Peter Winkler’s paper Seven Puzzles You Think You Must Not Have Heard Correctly.
Jan and Maria have fallen in love (via the internet) and Jan wishes to
mail her a ring. Unfortunately, they live in the country of Kleptopia
where anything sent through the mail will be stolen unless it is
enclosed in a padlocked box. Jan and Maria each have plenty of padlocks,
but none to which the other has a key. How can Jan get the ring safely
into Maria’s hands?
I don’t know whether this puzzle appeared before the Diffie-Hellman key
exchange was invented, but I am sure that one of them inspired the
other. The official solution is that Jan sends Maria a box with the ring
in it and one of his padlocks on it. Upon receipt Maria affixes her own
padlock to the box and mails it back with both padlocks on it. When Jan
gets it, he removes his padlock and sends the box back, locked only
with Maria’s padlock. As Maria has her own key, she can now open it.
My students suggested many other solutions. I wonder if some of them can be translated to cryptography.
- Jan can send the ring in a padlock box that is made of cardboard. Maria can just cut the cardboard with a knife.
- Jan can use the magic of the Internet to send Maria schematics of the
key so she can either 3d print it or get a professional to forge it. If they are
afraid of the schematics getting stolen Jan can send the schematics after the
package has been delivered.
- Jan can use a digital padlock and send the code using the Internet.
- Jan can send it in a secret puzzle box that can be opened without a key.
- Maria can smash the padlock with a hammer.
Now that we’ve looked at the Padlock Puzzle, let’s talk about
cryptography. I have an imaginary student named Charlie who doesn’t know
the Diffie-Hellman key exchange. Charlie decided that he can adapt the
padlock puzzle to help Alice send a secret message to Bob. Here’s what
Suppose the message is M. Alice converts it to binary. Then she creates a
random binary key A and XORs it with M. She sends the result, M XOR A,
to Bob. Then Bob creates his own random key B and XORs it with what he
receives and sends the result, M XOR A XOR B, back to Alice. Alice XORs
the result with her key to get M XOR A XOR B XOR A = M XOR B and sends
it to Bob. Bob XORs it with his key to decipher the message.
Each sent message is equivalent to a random string. Intercepting it is
not useful to an evil eavesdropper. The scheme is perfect. Or is it?
Here is a logic puzzle.
Puzzle. You are visiting an island where all people know each
other. The islanders are of two types:
truth-tellers who always tell the truth and liars who always lie. You
meet three islanders—Alice, Bob, and Charlie—and ask each of
them, “Of the two other islanders here, how many are truth-tellers?”
Alice replies, “Zero.”
Bob replies, “One.” What will Charlie’s reply be?
The solution proceeds as follows. Suppose Alice is a truth-teller. Then
Bob and Charlie are liars. In this situation Bob’s statement is true,
which is a contradiction. Hence, Alice is a liar. It follows, that there
is at least one truth-teller between Bob and Charlie. Suppose Bob is a
liar. Then the statement that there is one truth-teller between Alice
and Charlie is wrong. It follows that Charlie is a liar. We have a
contradiction again. Thus, Alice is a liar and Bob is a truth-teller.
From Bob’s statement, we know that Charlie must be a truth-teller. That
means, Charlie says “One.”
But here is another solution suggested by my students that uses meta
considerations. A truth-teller has only one possibility for the answer,
while a liar can choose between any numbers that are not true. Even if
we assume that the answer is only one of three numbers—0, 1, or 2—then
the liar still has two options for the answer. If Charlie is a liar,
there can’t be a unique answer to this puzzle. Thus, the puzzle
question implies that Charlie is a truth-teller. It follows that Alice
must be lying and Bob must be telling the truth. And the answer is the
same: Charlie says, “One.”
You might have noticed that my blogging slowed down significantly in the
last several months. I had mono: My brain was foggy, and I was tired
all the time. Now I am feeling better, and I am writing again. What
better way to get back to writing than to start with some jokes?
* * *
The wife of a math teacher threw him out from point A to point B.
* * *
At the job interview at Google.
—How did you hear about our company?
* * * (submitted by Sam Steingold)
50% of marriages end with divorce. The other 50% end with death.
* * *
People say that I am illogical. This is not so, though this is true.
* * *
Humanity invented the decimal system, because people have 10 fingers.
And they invented 32-bit computers, because people have 32 teeth.
* * *
When a person tells me, “I was never vaccinated, and, as you can see, I
am fine,” I reply, “I also want to hear the opinion of those who were
never vaccinated and died.”
* * *
I will live forever. I have collected a lot of data over the years, and
in all of the examples, it is always someone else who dies.
* * *
Just got my ticket to the Fibonacci convention! I hear this year is going to be as big as the last two years put together.
* * *
I am afraid to have children as one day I will have to help them with math.