This is an exam paper from the University of Cambridge it's part of a course which
is widely regarded as being the most demanding undergraduate mathematics course in Britain and
that is the Mathematical Tripos. The tripos comes in three parts spread over three years
and it is an old exam steeped in tradition and prestige. It was introduced around 1790
and still challenges some of the smartest students today. Some mathematicians would
like to see it abolished but for now the tripos still has a strong grip on English higher math
education so let's take a look at it. This is the exam that for over 200 years has been seen
as a leading test of mental agility and stamina. It's changed a lot in that time
but the current iteration still contains all sorts of really difficult questions. It starts
with number theory; compute the continued fraction expansion of the square root of 29.
Then an analysis proof then a cryptography
question asking one to describe the Berlekamp-Massey method for breaking
a cipher stream arising from a linear feedback shift register.
This is one of the four papers which a student would take in their third year of the tripos.
After this they can leave with a bachelor's degree or do one more year of study called part three to
obtain a masters of math. You'll notice though that this one exam paper contains questions on
everything from number theory to coding to statistics, quantum mechanics and analysis.
It’s more common elsewhere to see these different areas of math separated into
different modular exams where you only need to study for one at a time.
That’s not the case here, all the subjects are considered together but you do get to pick and
chose which questions to tackle. This approach allows students to attend whatever courses
and lectures they wish and then choose how they want to prepare for the exam.
Heres the list of courses on offer in third year. I’m drawn to see what exam
questions look like for some of these, for example math of machine learning.
This is the 2022 question they are asked to write down and adaboost algorithm,
to say what is meant by a convex hull and a convex combination of vectors,
what is meant by the empirical Rademacher complexity. Here's the syllabus for the
mathematics of machine learning course to see the other kind of ideas that go into this math.
Cambridge now has many triposes for several different courses including an english tripos
and natural sciences tripos but it used to be just math. The exam originally took the form of a
debate where the student had to wrangle logic but then became what might be the oldest university
written exam in the world. The original exam was called the Senate House exam because that’s
the name of the building where it was taken. The oldest copy of the exam that I could find
was this one from 1841. The questions are also difficult, but different
Solve the following equations and show that a quadratic equation cannot have more than two
roots. Prove that in a spherical triangle any one side is greater than the difference between
the two others. Enunciate and prove the tenth Lemma of Newton's first section. They expected
students to be familiar with Newton's work in a way that no student is today if I pull out
my handy Principia we might be able to get the answer to this one. Okay this looks like Lemma
10 in the first section; The spaces which a body describes by a finite force urging it
whether that force is determined and immutable or is continually augmented or continually diminished
are in the very beginning of the motion to each other as the squares of the times. hmm. The focus
on Newton does make sense since Newton did study in Cambridge and heavily influenced the British
math scene. Give the arguments for supposing the earth to be round and on the supposition
of its being a sphere investigate a formula for calculating approximately the dip of the horizon.
This older exam was a true mental agility and endurance test there are 12 papers to be taken
two a day for six days it's a week-long marathon answering hundreds of questions for five and a
half hours a day students trained for it like an Olympic sport. The highest scoring student
was called the Senior Wrangler and became a minor celebrity. I found the scores for an exam in the
1860s which gives an idea of scoring around this time and out of a possible mark of 17000
the senior Wrangler scored only about 7600, second place scored about 4100 and the lowest
scoring candidate to still pass scored just 237 points and earned the special Wooden Spoon award.
The 1841 results are listed here with Stokes the Senior Wrangler. That is George Stokes
now famous for the Navier-Stokes equation in fluid mechanics and Stokes' theorem in vector
calculus. You might have been examined on those ideas yourself and now you've seen the exact exam
questions Stokes faced as a student. Let's compare some similar questions from then and now. Given
that this exam took place before Einstein's work on relativity we can't expect much of an analog
to this general relativity question but there are plenty of questions about Newtonian gravity. 1841
was before we developed astronomical spectroscopy or measured the expansion of the universe so this
cosmology question would blow the mind of even an 1841 Senior Wrangler- but wait actually George
Stokes apparently is credited with inventing the fundamental principles of spectroscopy.
Modern quantum mechanics wasn't to be invented for another 60 years so we won't see that but
there are plenty of questions about light and we won't find much machine learning,
statistical modeling or coding and cryptography in the 1841 paper since there were no computers.
These are the best I could find to resemble the modern number theory and analysis questions.
Training for the tripos has historically involved external coaching to help students
increase the speed that they can solve and write out solutions and the whole process
has been criticized for not allowing enough time for a deep understanding or to make
investigations which are important skills for mathematician and the Senior Wrangler
has not always turned out to be the most distinguished mathematician later in life.
One notable critic of the tripos was G H Hardy, Ramanujan's mentor, Hardy gave an address to the
mathematical Association in 1926 calling to abolish the tripos completely he said that
the ability to show originality in mathematics can't be tested under exam conditions and so the
exam reduces to a kind of game. A student can choose to play the game with all their force
but must realize they are putting their mathematical development on hold to do
so. The prestige and status of the tripos forces many students to devote everything to the game.
Hardy isn't against exams in general and knows they are useful to test students but he wants
exams that do not profess to be more than a reasonable test of certain humdrum qualities.
The qualities worth testing on an exam would carry any student at Cambridge to second class
and beyond that he does not believe in recognizing differences of ability by examination. The tripos
is an exam that attempted to sort students right to the top, today students are no longer publicly
ranked in order but the Senior Wrangler is still acknowledged during the reading
of results at Senate House. Students today might still relate to the idea of having to put their
real mathematical pursuits on hold in order to play the game. Hardy says if Einstein had
taken the tripos what would it matter what place he took? Students are driven to perform in the
exams by tutors and greedy college authorities. "I do not want to reform the tripos but destroy it,
but neither I nor anyone else will succeed in doing so since the practical difficulties would
be so serious and the force of tradition is so terribly strong." Some things have changed
since 1926. Hardy's view is that the tripos should produce research mathematicians but not everyone
studying math has that aim. The modern day tripos website is proud that the course prepares students
for a range of jobs not just academia. Let me show you one more interesting past paper,
this was the first page of the tripos exam in 1890 and was the exam taken by Philippa Fawcett. She
wasn't the first woman to sit the math tripos but was the first to rank number one. As the results
of the exam were announced the men's results were read first and then the women's. Women weren't
officially ranked but were placed relative to the men. Fawcett was placed above the Senior Wrangler
and Senate House erupted in celebration. She beat the Senior Wrangler by 400 marks and her success
was headline news. It took another 58 years before women were actually allowed to receive
degrees from Cambridge. Today the results are read out in alphabetical order rather than by rank,
but what do you think does the tripos deserve to be destroyed as Hardy wanted or is it a worthy
challenge that motivates students to perform at their best? Thanks for watching and a special
thank you to my Patreon supporters including today's Patron cats of the day Pan and Tilt.